How to prove that a matrix is invertible $\iff$ invertible at right $\iff$ invertible at left?
If $AB = I$ then $BA = I$
Some mesh-edges in object-mode aren't displayed So I have been extruding a cube to make a low poly banana and when I go from edit to object mode half the edges don't appear as well as half is dark.      I tried looking for an solution couldn't find one and its making me feel like I am doing something wrong so I decided to come here. How does one write on the outside of a circle?  I am looking for sample files/packages which imitate a magazine lay-out. Specifically a page with two columns having a circular picture between the two columns.    Example of the desired layout, created using Word: How do I install and manage RPMs? Let $\{x_n\}$ and $\{y_n\}$ be sequences of real numbers which converge to $\ell$ and $m$ respectively. Show that  $$\lim_{n\to \infty} \frac{1}{n}\sum_{k=1}^{n}x_ky_{n+1-k}=\ell m$$  This is a problem I found in a book, it seemed to be routine, but now I can't solve it.  What I tried :     Let us take some $\epsilon&gt;0$. Now after a stage $N$, both $|x_n-\ell|&lt;\epsilon$ and $|y_n-m|&lt;\epsilon$. Now I wanted to break the sum $$\left|\frac{1}{n}\sum_{k=1}^{n}x_ky_{n+1-k}-\ell m\right|$$in two parts, one upto $N$, and other one for $N+1$ to $n$. Now we need this to be arbitrarily small. But here I am lost. I don't know how to proceed anymore. Because when I take $x_i's$ to make up a finite sum, then the $y_j's$ are there too with $j$ very large. I think this is the main idea, but unable to do anything further. Can someone help me? Thanks. Number of ways to connect sets of $k$ dots in a perfect $n$-gon Two logos in beamer after certain frame I am making a presentation which is devided in two parts. The first part is about experiments held in instute A and the second part is about experiments held in instute A and institute B.  Is it possible to add a second logo in a beamer presentation after a certain frame?  Let's say that this is a previous slide    This is a next slide Evaluating a mathematical expression in a string Why are there no animals in "The Matrix"? Aside from the repetitive appearance of the black cat, I can't remember seeing any living animals, either inside the Matrix simulation or in Zion.  How come? Is there an in-universe explanation? Let me start this with : Very long "About me" text squeezes both the left side (flair) and the right side (the section with location, GitHub link, etc.)  Now, the reverse thing has happened:         put a long text in their "location", and it causes the right section to be overly extended, effectively squeezing the About Me section.  When will the squeezing game end? How many ancestors alive 1000 years ago does a person have on average? I was thinking about something the other day (that is probably stupid), and thought this would be a good place to talk about it.  If someone wanted to draw its family tree over the last 1000 years, he'd write a binary tree with a depth of about 40 (counting 25 years on average between each generations). This tree would have 240 leaves, or about 1012 ancestors on the deeper level. That is obviously much more than the total world population and a stupid figure when you think about it.  The only explanation that I can think of is that a real genealogy tree must have "linked" branches at several levels that change the actual number of ancestors. Anyway, for an average person, how many ancestors would that make? Could one assume that all or nearly all of the persons that lived in its country or world region a 1000 years ago would be part of its ancestors? Query to compare the structure of two tables in MySQL To automate the backup process of one of my MySQL databases, I would like to compare the structure of two tables (current version vs old version).  Can you think of a query that can compare two tables?  Here are some example tables that you can compare.   CREATE TABLE product_today (   pname VARCHAR(150),   price int,   PRIMARY KEY (pname) );  CREATE TABLE product_yesterday (   pname VARCHAR(150),   price int,   PRIMARY KEY (pname) );  CREATE TABLE product_2days_back (   pname VARCHAR(15),   price int,   PRIMARY KEY (pname) );   The first two tables have identical structures. The last one is different. I just need to know whether two tables have different structures or not. I'm not interested in the how they differ. How can I know if a branch has been already merged into master? I have a git repository with multiple branches.   How can I know which branches are already merged into the master branch? Programmatically disable apt unattended upgrades How can I programmatically/script disable apt unattended upgrades? Running Ubuntu 16.04. How to get the IPV4 address for an interface from /proc Is there a clean, simple way to get an IP address for a network interface from /proc, similar to the way I can get the MAC address for a network interface? Ideally I would just type cat /proc/&lt;foo&gt;/{interface_name} and get the IPv4 address.  I'd rather not run anything other than cat. SVM for unbalanced data I want to attempt to use Support Vector Machines (SVMs) on my dataset. Before I attempt the problem though, I was warned that SVMs dont perform well on extremely unbalanced data. In my case, I can have as much as 95-98% 0's and 2-5% 1's.   I tried to find resources which talked about using SVMs on sparse/unbalanced data, but all I could find was 'sparseSVMs' (which use a small amount of support vectors).  I was hoping someone could briefly explain:   How well SVM would be expected to do with such a dataset   Which, if any, modifications must be done to the SVM algorithm What resources/papers discuss this I am trying to compile a list of the different ways to come up with the closed form for the sum in the title. So far I have the famous Gauss story, the argument by counting doubletons, and using generating functions. Is there any other (significantly) different way to discover the closed form?  To clarify, Gauss's method was to take $1+2+3+\dots+n$, write it backwards, and add it up. I switch back and forth between two computers constantly:   OSX Snow Leopard  Ubuntu 10.10    I'd like to be able to make Ubuntu use the same keyboard shortcuts as OSX System Wide, for the following keyboard sequences only (ie I don't want to make Ctrl act like Super)   Super+C -> Copy Super+V -> Paste Super+T -> Open a new tab in whatever browser I'm in.  Super+W -> Close a tab in whatever browser I'm in.    So in short, is there anyway for me to map just these keyboard sequences to the following, system wide ?   Super+C -> Ctrl+C Super+V -> Ctrl+V Super+T -> Ctrl+T  Super+W -> Ctrl+W   I know there are ways for me to do this for vim, and Firefox, and I'm sure specifically for most applications... but I would prefer to have to do this just once and have it work that way system wide!  I'm using a standard PC keyboard, that is "Generic 105 key (intl) PC" on Ubuntu. I'm also using the same keyboard on my Mac mini. How to discard unsaved changes to a view when editing it leads to a WSOD? How to create a Wi-Fi Hotspot (in Access Point mode)? Group algebra ever a field? Let $G$ be a group. Let $K$ be any field and consider its group algebra $K[G]$. It is known that if $G$ is a finite group then $K[G]$ is never a field. Or even if $G$ is not finite: if $G$ has at least one non-trivial torsion element, then $K[G]$ is not a field.  Question: Do there exist infinite groups $G$ such that $K[G]$ is a field or a division ring?