Problem: a) Determine which amounts of postage can be formed using just 4 cent stamps and 11 cent stamps. b) Prove your answers to a using strong induction. My work: (I am only working on part a for now)  I saw that this problem was a variation of this one  but that one came with the initial bounds, &quot;Prove that every amount of postage of 12 cents or more&quot;, but here, you have to come with the initial bounds.  I watched a Youtube video on how to do that problem  and here's the author's basis case  Here's my work off that author's basis case 12 = 3(4) + 0(11) 19 = 2(4) + 1(11) 26 = 1(4) + 2(11) 33 = 0(4) + 3(11) And my conclusion is that postage of 12 + multiples of 7 can be formed with just using 4 cent stamps and 11 cent stamps. Is that the correct conclusion to which amounts of postage can be formed using just 4 cent and 11 cent stamps? I wasn't completely sure because 4 can be formed with just using 1 cent stamp but I felt like there was a reason you start from 12 in the other problem, to make the induction step easier? Would you have to include numbers like 4 and 11?
I want to determine the set of natural numbers that can be expressed as the sum of some non-negative number of 3s and 5s.  $$S=\{3k+5j∣k,j∈\mathbb{N}∪\{0\}\}$$  I want to check whether that would be: 0,3, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and so on.  Meaning that it would include 0, 3, 5, 8. Then from 9 and on, every Natural Number. But how would I explain it as a set? or prove that these are the numbers in the set?
Proof of matrix norm property: submultiplicativity I've been searching for the definition of the submultiplicative (I think it has multiple names from what I've seen) property in proof form. Some books define it as part of the properties that define matrix norms, and some include it as an additional property. I still haven't been able to work it out for myself or find it anywhere.  Let $A$ and $B$ be $n\times m$ and $m\times l$ matrices respectively, prove that:  $$\begin{align} \|AB\| \le \|A\|\|B\|  \end{align}$$ Restoring open software after a restart event in windows I find that at the end of a long day, I sometimes have a large number of programs running. All which I will need to use tomorrow. Normally, this isn't an issue, I can simply lock the machine and come back tomorrow. My problem arrises when windows update launches in the middle of the night and force restarts my computer. That in turns closes all my open software. I of course save everything regularly so I don't loose anything, but I waste time reopening all of those resources whenever there is a restart.   [EDIT]  I should clarify that I still want to be able to restart my computer when an update comes down. Preventing the restart only delays the problem until later. I should have been more specific in that I want to be able to recover my working environment after a restart for any reason. Things like scheduled maintence, power loss, updates, and software installs.  [EDIT]  I can't simply have them setup to launch at startup becasuse those files change from week to week. So I need something that monitors what I have open, and gives me the option to "recover" those software sessions when I log back in.  Anyone have any suggestions on what I can do? I'd even be willing to purchase software to do this for me if that is the only option.  Thanks "Add" button doesn't work with JavaScript off I'm using stackoverflow.com with JavaScript off. I accept this comes with somewhat less functionality, however for the most part SO works fine.  If I look at the home page (), every single button on there except for two types works with JS off.  The first second is that the "x Votes; y answers;z views" mouse-over indicates they're links but clicking them does nothing. This one is less of an issue because the user can still click the question title, and they're only implied buttons anyway.  The second (and bigger) issue is the "Add" buttons next to "Favorite Tags" and "Ignored Tags". These are explicit buttons that don't do anything with JS off. There is no working JS substitute for them (the "tag subscriptions" link itself works, but the page it goes to doesn't work with JS off, but that's another bug report). Need a matrix to seventh power which is the identity but the original is not or the negative identity The question on the homework wants a $2\times2$ matrix $R$, but $R$ not equal to positive or negative identity, that when you raise to seventh power you get the identity.  $R^7=\operatorname{Id}$.  I thought it would have to be some ones and negative ones,  it no luck, then tried some ones and some $.5$.  Just using an online matrix calculator to do brute force.  We just started talking about eigenvalues with visual example, so assuming wouldn't need that since we have not talked about how to find them yet.  Any hints or ideas to get me back on track? Is it possible to flash a custom ROM without a custom recovery image For an un-rooted device, almost all articles/forum posts I've read say that in order to flash a custom ROM you first need to flash (or boot) a custom recovery image like TWRP.   I'm not sure I understand why. If the bootloader is unlocked shouldn't it be possible to manually flash images that make up the custom ROM to the required partitions with fastboot flash &lt;partition&gt; [ &lt;xxxx.img&gt; ], and leave the stock recovery untouched.  I realize that this may be difficult and that there is no benefit in doing it this way, but I'm curious to know whether or not it is possible, and if not, why? Both myself and my boyfriend have PS4s.  Can I just get a single PSN account and sign both consoles into the same account, or do I need to get two separate accounts? How do I manage how many revisions snapd keeps of an application? When quoting a quotation, how do you handle the double quotes? Assume there is original source text:     This restaurant is amazing with "delicious lasagna" and great service.   Make sure you go 6-7pm for the quietest times.   I want to quote "delicious lasagna" but not sure how to handle the outer double quotes.  If I was quoting more than just that quotation (e.g. is amazing with "delicious lasagna" and great service) I'd just do:     "...is amazing with 'delicious lasagna' and great service."   However, when quoting just that quotation, it looks like:     "'delicious lasagna'"   Is this the correct way or is there something better? Say you have the infinitely continued fraction: $$x+\cfrac{1}{x^2+\cfrac{1}{x^3+\cfrac{1}{x^4+\ddots}}}$$ When $x=1$, you can see that it's $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$ which converges upon the golden ratio $\phi = 1.61803398875...$. What if I were to plug in $x=2$? $$2+\cfrac{1}{4+\cfrac{1}{8+\cfrac{1}{16+\ddots}}}$$ Using a caluclator, it seems to converge upon the irrational number: $2.24248109286...$ Is there any way I can represent this irrational number algebraically? Unknown site on Android app When I select the unknown site (i.e. the meta sites) the app crashes.   I have a Samsung Note 5 and version 1.0.89 of the app. Allow setuid on shell scripts The setuid permission bit tells Linux to run a program with the effective user id of the owner instead of the executor:  &gt; cat setuid-test.c  #include &lt;stdio.h&gt; #include &lt;unistd.h&gt;  int main(int argc, char** argv) {     printf("%d", geteuid());     return 0; }  &gt; gcc -o setuid-test setuid-test.c &gt; ./setuid-test  1000  &gt; sudo chown nobody ./setuid-test; sudo chmod +s ./setuid-test &gt; ./setuid-test  65534   However, this only applies to executables; shell scripts ignore the setuid bit:  &gt; cat setuid-test2  #!/bin/bash id -u  &gt; ./setuid-test2  1000  &gt; sudo chown nobody ./setuid-test2; sudo chmod +s ./setuid-test2 &gt; ./setuid-test2  1000     Wikipedia :     Due to the increased likelihood of security flaws, many operating systems ignore the setuid attribute when applied to executable shell scripts.   Assuming I'm willing to accept those risks, is there any way to tell Linux to treat the setuid bit the same on shell scripts as it does on executables?  If not, is there a common workaround for this problem? My current solution is to add a sudoers entry to allow ALL to run a given script as the user I want it run as, with NOPASSWD to avoid the password prompt. The main downsides to that is the need for a sudoers entry every time I want to do this, and the need for the caller to sudo some-script instead of just some-script Problem. Use Tutte's 1-factor theorem to prove that every 3-regular graph with no bridges has a 1-factor.  I honestly have no idea how to approach this problem. I have drawn a couple of graphs to gain some insight to no avail. Any hints or tips will be greatly appreciated. bash: /usr/bin/locale-check: No such file or directory error after i downgraded from ubuntu 18.10 to 18.04 I have been trying for a long time to change Blender's icon, but no matter what I do, I am not allowed to edit anything in /snap/blender-tpaw/3/.  Here's what I tried:   Editing the files from nautilus without sudo. Editing the files from nautilus with sudo (sudo nautilus in terminal ). Using terminal commands such as cp or rm without sudo. Using terminal commands with sudo (such as sudo cp &lt;source&gt; /snap/blender-tpaw/3/ or sudo rm /snap/blender-tpaw/3/&lt;filename.ext&gt;) Doing everything above in a root terminal (using sudo -i)   In every case I get the following error:  cannot remove/copy '/snap/blender-tpaw/3/filename.ext': Read-only file system   where filename is the file and .ext is its extension.  This also applies to other snaps' files, not only Blender.  Am I doing something wrong here? Or is it just impossible to change those files? Although I don't think it is impossible because everything here from Ubuntu to Blender is open-source, so they have no reason to block us from modifying those files.  EDIT:  I used Main Menu (alacarte) to change the icon, but I still want to know why I cannot modify any snap file. Automatically redirect anonymous user from unanswered duplicate question to corresponding answered version How do you add a certificate authority (CA) to Ubuntu? Some of my Apps Icons are changing to a generic app icon, and I have no idea why. Tried to restart the system and nothing. 3 days ago i updated my Skype and then the icon is gone. Subjunctive mood: 'was' usage after 'I wish' and 'if only' had declared the sentence below as follow: 1) STAR TURN, 2) BEST LINES.      I wish Bernard Cribbins was my grandad. Perhaps he would be willing to   adopt?     Rattigan: “If only that was possible.” The Doctor: “If only that were   possible. Conditional clause.”    Is 'was' after 'I wish', 'if only' and 'if' (in conditional clauses) colloquial?  Does it be used in formal speech and writing? How can I do case insensitive string comparison in Python?  I would like to encapsulate comparison of a regular strings to a repository string using in a very simple and Pythonic way. I also would like to have ability to look up values in a dict hashed by strings using regular python strings. If I multiclass into 2 or more classes/subclasses that can cast spells, how do I determine what spells I know or can prepare?  For instance, if I am playing a multiclassed character with 1 level in warlock, 2 levels in bard, and 3 levels in paladin, how do I determine which spells I know/have prepared?    Related: