Let $A$ and $B$ be two $n$-by-$n$ real matrices such that $A+B = AB$.   How do I prove that $AB= BA$?   I have tried using the trace function on $A+B-AB$. But I could not get any Ideas. Kindly provide me with hints.
$A$ and $B$ are $n\times n$ matrices and $A+B=AB$. I have an interesting proof that this implies $A$ commutes with $B$, but the proof only works when $||B|| \lt 1$. I'm looking for a way to salvage the proof so that it works for all $B$.  $$A=AB-B$$ $$A=(A-I)B$$ by repeatedly substituting $A$ with $(A-I)B$ on the RHS, $$A= (A-I)B^N - \sum_{i=1} ^{N-1} B^i$$  Since the space of $n\times n$ matrices is Banach (hence complete) under the standard operator norm, the limit $\sum_{i=1}^{\infty}B^i$ is well defined given $||B|| \lt 1$. Hence, taking the limit as $N \rightarrow \infty$, $$A =  -\sum_{i=1}^{\infty}B^i$$  Now the LHS commutes with B, hence so does the RHS.    Is there some trick to extend the method to work with any $B$ where $||B|| \geq 1$? I know that the result is true due to another simple algebraic proof, but I'd like to see if this method can be salvaged.
How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$? I've spent the better part of this day trying to show from first principles that this sequence tends to 1. Could anyone give me an idea of how I can approach this problem?  $$ \lim_{n \to +\infty} n^{\frac{1}{n}} $$ How to calculate $(G/H)/(K/H)$? I have a group $G=\mathbb{Z}_{36}$ with normal subgroups $K$ and $H$, where $H$ is also a normal subgroup of $K$. What I'm really supposed to find is a bijection between $G/K$ and $(G/H)/(K/H)$ by using the third isomorphism theorem. I have already calculated $G/K$, $G/H$ and $K/H$, but I do not really know how to calculate $(G/H)/(K/H)$. Say that one of the elements in $G/H$ is $\{0,18\}$ and one of the elements in $K/H$ is $\{0,18\}$. Will one of the elements in $(G/H)/(K/H)$ be as follows? $$\{0,18\}+\{0,18\} = \{0+0,18+18\}=\{0,36\}=\{0\}$$ Another way I'm thinking it's possible to caluclate is $$\{0,18\}+\{0,18\} = \{0+0,0+18,18+0,18+18\}=\{0,18,18,36\}=\{0,18\}$$ Is any of these a correct way of calculating an element in $(G/H)/(K/H)$? SQL Server equivalent to Oracle's CREATE OR REPLACE VIEW In Oracle, I can re-create a view with a single statement, as shown here:  CREATE OR REPLACE VIEW MY_VIEW AS SELECT SOME_FIELD FROM SOME_TABLE WHERE SOME_CONDITIONS   As the syntax implies, this will drop the old view and re-create it with whatever definition I've given.  Is there an equivalent in MSSQL (SQL Server 2005 or later) that will do the same thing? How to keep Heading together with text? I use a very custom style (not mine) and very frequently I get a heading  on the bottom of a page and the next paragraph on the top of the next page. Sometimes this paragraph may be a list or similar. The same is true for listings and the listing description.   Is there a general way how I can adjust the "badness" of these events? Or can I adjust when a section heading would be on the last, say, 3cm of a page it should go to the next page? Or can I say, if a *listing description" is on the top 3cm of a page pull 5 more listing lines from the last page onto this page?   The style I use does not use standard structure tags and environments. But lets call them section, subsection, listing and I will try to migrate them to my lingo. Why do researchers need universities? This is a follow-up question to  and probably an even more naive question. If I'm understanding the answer correct, professors need funding to do research, but once they get it, the university takes some of the grant as overheads (to pay for office space, electricity, etc), and the cut the university takes is substantial.  Given that then, why do professors need universities? One could just apply for the grant as per normal, and once one gets it, buy a slightly bigger house and convert one of the rooms into a lab. One loses nothing to overheads, gets to work from home, has zero teaching duties, can choose to settle anywhere (no two-body problem!), and can even monitor an experiment 24/7. This gets even easier if one works in a field that doesn't need a physical lab. Further, presumably the grant covers postdoc salaries, so one would still be able to pay for postdocs (although probably not PhD students since a professor without a university will not be able to award a degree).   The obvious answer is that one cannot apply for a grant without a university, but Google indicates that's not the case, e.g.  only looks at one's previous accomplishments &amp; potential applications of the research result, both of which are independent of the university. I suppose one could lose journal access, but there's always stuff like arXiv / ResearchGate, emailing the authors of the desired paper, or even Scihub (oops). It's conceivable that not working for a university loses one some prestige, since one can no longer claim to be a professor. However even then I'd expect at least some academics to choose this path, valuing the convenience &amp; extra research funding over prestige.  If the answer to this varies from field to field, I'm most interested in the sciences. Which is better: Employer-sponsored HSA or opening an HSA individually? So I know that:   Employer-sponsored HSA's allow you to deduct contributions directly from your paycheck, pre-tax With individual HSAs, you must contribute using post-tax dollars, then deduct those contributions from your taxable income when you do your taxes   Given those facts, I guess it's marginally better to use an employer-sponsored HSA because it means the government doesn't temporarily hold on to your taxed income, only to return it later after you do your taxes; you could theoretically put that money to work in an investment vehicle instead of having the government hold it hostage for a while.  But are there any other benefits (either way) that I'm not aware of? The  kind of threw me with the following statement:     The main advantage of making pre-tax contributions is the Federal Insurance Contributions Act tax (FICA) and Medicare Tax deduction, which amounts to a savings of 7.65% each to the employer and employee (subject to limits of the Social Security Wage Base).   But I'm not really sure what that's telling me.  My employer is offering an HSA, but is not offering to contribute to the HSA or match my contributions. So if it's all pretty much the same, I'd rather use the bank of my choosing and manage it privately.  Thanks. Zombies - Why do they need to feed on living animals/humans? Why do only two sexes exist for animals? Pigeonhole-principle with two choices I am able to solve this sort of problem pretty easily.     An arm wrestler is the champion for a period of 75 hours. The arm   wrestler had at least one match an hour, but no more than 125 total   matches. Show that there is a period of consecutive hours during which   the arm wrestler had exactly 24 matches.   However this homework problem I can't seem to figure out.     Tyler has five weeks to prepare for his driver’s test. His mother   volunteers to ride with him for either 15 minutes or one half hour   every day until the test but not for more than 15 hours in all. Show   that during some period of consecutive days, Tyler and his mother will   drive for exactly 8.75 hours.   If I try doing it the same way as the first problem. I get $70$ pigeons and $95$ pigeon holes. I honestly don't know where to start with this problem. I thought that maybe I could assume that they only drove $.25$ hours on each day and prove that there is a period of consecutive days where they drove for exactly $8.75$ hours. Then I could say that we assume there is one day that they drove for $.5$ hours and work my way up, but I can't seem to get that to work either. When flagging, there are two special case options: spam and rude or abusive.   What is spam, and when should I flag content as such? What is considered offensive, rude or abusive content? How does the spam flag differ from the rude or abusive flag? What is the effect of these special flags? When are these flags removed? How can I upgrade from Mac OS X Tiger 10.4? If I buy a regular retail Snow Leopard disc (i.e. the £25 version), can I use it to upgrade a Mac from Tiger (10.4)?   on the Apple Store’s Q&amp;A site suggests it’ll work, but Apple seem to imply that you need the (significantly more expensive) Mac Box Set (“”). Hydrogen radial wave function infinity at $r=0$ Getting below error message while issuing :  sudo apt-get update  Get:1 http://us.archive.ubuntu.com/ubuntu xenial InRelease [95.8 kB] Ign:2 http://repo.mongodb.org/apt/debian wheezy/mongodb-org/3.2 InRelease                                                                                                             Ign:3 http://dl.google.com/linux/chrome/deb stable InRelease                                                                                                                          Hit:4 http://ppa.launchpad.net/canonical-x/vulkan/ubuntu xenial InRelease                                                                                                 Hit:5 http://repo.mongodb.org/apt/debian wheezy/mongodb-org/3.2 Release                                                                             Hit:6 http://us.archive.ubuntu.com/ubuntu xenial-security InRelease                                                   Ign:7 http://dl.google.com/linux/talkplugin/deb stable InRelease                                                      Hit:8 http://us.archive.ubuntu.com/ubuntu xenial-updates InRelease                                                    Ign:9 http://linux.dropbox.com/ubuntu wily InRelease                                            Hit:10 http://ppa.launchpad.net/numix/ppa/ubuntu xenial InRelease                               Get:12 http://dl.google.com/linux/chrome/deb stable Release [782 B]       Hit:13 http://dl.google.com/linux/talkplugin/deb stable Release                   Ign:14 https://apt.dockerproject.org/repo ubuntu-wily InRelease           Hit:15 https://apt.dockerproject.org/repo ubuntu-wily Release             Get:16 http://dl.google.com/linux/chrome/deb stable Release.gpg [181 B]   Hit:17 http://linux.dropbox.com/ubuntu wily Release                                   Get:20 http://dl.google.com/linux/chrome/deb stable/main amd64 Packages [1,191 B] Fetched 98.0 kB in 0s (118 kB/s)                                  Reading package lists... Done N: Skipping acquire of configured file 'main/binary-i386/Packages' as repository 'http://dl.google.com/linux/chrome/deb stable InRelease' doesn't support architecture 'i386'   This cuts across various releases of ubuntu How do you declare an interface in C++? Accessing class members on a NULL pointer I was experimenting with C++ and found the below code as very strange.   class Foo{ public:     virtual void say_virtual_hi(){         std::cout &lt;&lt; "Virtual Hi";     }      void say_hi()     {         std::cout &lt;&lt; "Hi";     } };  int main(int argc, char** argv) {     Foo* foo = 0;     foo-&gt;say_hi(); // works well     foo-&gt;say_virtual_hi(); // will crash the app     return 0; }   I know that the virtual method call crashes because it requires a vtable lookup and can only work with valid objects.   I have the following questions    How does the non virtual method say_hi work on a NULL pointer? Where does the object foo get allocated?   Any thoughts? I'm having trouble with a proof by induction Structure of groups of order $pq$, where $p,q$ are distinct primes. I don't know about the Sylow Theorems.  But I have been wondering about a proof of the fact that a group or order $pq$ where $p$ and $q$ are distinct primes must be cyclic. I can't quite work out the details, but here is the general idea. I would like help with filling in details. I assume that it is already known that $G$ has subgroup(s) of order $p$ and subgroup(s) of order $q$.  If $G$ is a group of order $pq$ ($p\neq q$), then I know that $G$ has a subgroup $H$ of order $p$ and a subgroup $K$ of order $q$. Then $H\simeq \mathbb{Z}_p$ and $K\simeq \mathbb{Z}_q$.   But then $H\oplus K \simeq \mathbb{Z}_{pq}$, so I would think that $H\oplus K \simeq G$. I guess one could do an internal direct product instead of an external direct product, but I don't know that $H$ and $K$ are normal subgroups.  I am asking for help completing this argument.  Edit: I see from the comments below that I might need to assume that the smaller prime does not divide the larger prime minus $1$. Or maybe it is enough to assume that the primes are greater than or equal to $3$ (Still distinct). I have to design a circuit in which I have to transfer data through lasers (serial transmission of data). Now what I was thinking was to check if the laser light is on then '1' is passed and when the laser is off '0' would be passed. Now I need to check if a 1 or a 0 was passed.   How would I check for this and another problem I'm facing is if it's on it may keep on passing 1's. How would I check for this? I though of using LDR's (Light Dependent Resistors)? Would this be okay or it there a better method? What is the correct vector input for a texture connected to a Cycles lamp so that I can make a projector-like textured light?  I am trying to make an projector-like effect with a textured light in Cycles, currently trying it out with a Spot type of lamp, but I am open to any solution involving other lamp types if they work correctly.  The problem is that whatever vector input I use, the texture is always projected downwards, towards the global $-Z$ axis in scene world space, no matter where I point my spot light. I would like to have the texture always project "forward" towards the the direction the spot light is pointing, in the local $-Z$ axis.    The plan would then be to use this spot at several places in my scene, pointing at different directions, and as it currently stands I would have to have a different node setup for each spot instance, manually compensating for each unique rotation in the scene, which would quickly become unpractical.  I have searched around the internet and I am currently using a solution similar to the one on this .  Is there any better method, perhaps involving some funky Vector Math, or is this not currently supported? What should I opt for?   Option A) very well paid PhD, around €3000 per month at a respectable university or;  Option B) a non-paid PhD, where I have to pay €2000 per year at one of the top 10 universities of the world?  EDIT: Actually they are both UK universities, Option B  is ranked between the top 10 in the world in engineering and tech, and I do get a scholarship that pays 18000 pounds which covers the difference between international student tuition and EU tuition. In Option A, I will be doing research in a good university and will be working alongside giants in the industry.