Let $R$ be a unitary ring and let $a,b \in R$ such that $ab=1$.  Does it imply that $a$ is invertible?  The definition of invertible element requires that $ab = ba = 1$, so I guess it doesn't imply that a is invertible, but I didn't manage to find a unitary ring in which this happens.
Can I have a hint on how to construct a ring $A$ such that there are $a, b \in A$ for which $ab = 1$ but $ba \neq 1$, please? It seems that square matrices over a field are out of question because of the determinants, and that implies that no faithful finite-dimensional representation must exist, and my imagination seems to have given up on me :)
I remember seeing an upper bound for the binomial $\binom{n}{k}$ with an exponential function, something like $\binom{n}{k}\leq \left(ne/k\right)^k$. What exactly is it, and are there other similar good upper bounds for $\binom{n}{k}$?  Edit: As the link in Macavity's comment shows, the bound is indeed $\binom{n}{k}\leq \left(ne/k\right)^k$. How can we prove this? Can you use external money to pay trading commissions in tax-free and tax-deferred accounts? Is there a drupal.org API for retrieving the list of available modules, a description of them and available versions? Change style properties of several layers simultaneously? XSL That Returns the XML unchanged Merging two incoming broadband lines for faster Internet We have a broadband line coming into our building which delivers around 8-10Mbps, which is okay but not great. Fiber is available in our area (30-100Mbps) but it's not available where we are just yet.   Is there some way we could connect two lines (which we would take from two different providers so we would have redundancy if one failed) and merge them into one signal which we could distribute?  Is there some sort of setup that can do this?  Of course we have the option of a leased line but for 12Mbps symmetrical upload/download we'd be looking at around £500 per month compared to £15 per month for normal or fiber broadband.  I made a small pseudo flow chat of the setup below: I have a material assigned to a few test objects with an image sequence as a texture. If I view the scene in render preview, and scrub the timeline, (or render a sequence,) the texture behaves as expected.  However, with the viewport set to 'texture' or 'material' I only see the first frame of the sequence, wherever I am on the timeline. 'Auto Refresh' is set in the cycles texture node. Is this the expected behaviour? How can I use the MS JDBC driver with MS SQL Server 2008 Express? My configuration:     windows XP SP3 JDBC 2005 MS SQL Server 2008 Express, exposed via tcp/ip on port 1433 sqljdbc.jar in class path    I tried:  try {     Class.forName("com.microsoft.sqlserver.jdbc.SQLServerDriver").newInstance();     con = DriverManager.getConnection("jdbc:microsoft:sqlserver://localhost:1433/SQLEXPRESS2008;databaseName=Test;selectMethod=cursor", "sa", ""); } catch (Exception e) {     e.printStackTrace(); }   But it always throws an exception:  java.sql.SQLException: No suitable driver   I also tried the following urls:  localhost:1433/SQLEXPRESS2008  localhost/SQLEXPRESS2008  localhost   Same results.  Any help? Why does blowing on someone who is wet feel colder than on someone who is dry? Reference: Why are my "special" Unicode characters encoded weird using json_encode? I have a Dirichlet distribution from which I'm maximizing $\alpha$ by calculating the log likelihood with the following equation  $p\left(s|\alpha\right) = \sum\limits_{i=1}^O\log\left(\frac{n_i!}{\prod\limits_{j=1}^kn_{ij}!}\frac{\Gamma\left(\alpha_0\right)}{\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)} \prod\limits_{j=1}^k \frac{\Gamma\left(n_{ij}+\alpha_j\right)}{\Gamma\left(\alpha_j\right)}\right)$  I have a python script to calculate the log likelihood. It works for small values, but when $n_{ij}$ gets high $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ python runs into overflow error (the max for math.gamma is between 171 and 172).   Is there a way to change the likelihood function so that $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ don't run into overflow errors. Can I just divide the values in $\Gamma$ by for example 100, if I then compare the likelihood between $p(s|\alpha)$ with different $\alpha$ values do the best $\alpha$ values still have the highest likelihood? How do I scale the enormously sized desktop icons on Ubuntu 18.04? Finite Summation of Fractional Factorial Series finding the index of a max value in R How might the weight of a falling object affect the damage it causes? Fall damage is 1d6 per 10 feet. What adjustments if any should I make for objects falling on a player character? (e.g. a bear)  Assuming the objects are meaningful threats but not instant character death, should the weight of an object change the calculation, e.g. more then 1d6 per 10 feet.  Or is this more in the spirit of improvising damage chart? i.e. the setback (cat to face) dangerous (orc fell on me), and deadly (the large bear).  If this is house-rule territory does any one have any experience or advice beyond  relevant to 5e. Import a module from a relative path When handling multiple white-on-transparent png images, I am hindered by the fact that the thumbnails of these images are all-white. Both Adobe Bridge CS6 and Windows Explorer suffer the same problem, I can't discern between two of these pngs.  Is there a way to make these thumbnails useful? Is there some setting I missed that allows changing the thumbnail background colour for a png -- now a deafult white? In IE 9 you can type in whatever number you want to get the perfect zoom but in Chrome all I see is the + and - in the drop menu.  How do I choose a custom zoom and set it as the default? Jawa asked me this  and I didn't know the answer. So I present it to the meta SO community:     [To get the Yearling badge] Do you need to earn the 200 rep within that one year or is it enough to get 200 at some point after being a member for a year? And does 'a year' mean the very first 12 months or the last (sliding window of) 12 months?    I would assume that if you hit 200 reputation after you've already been around for a year, then you would get the Yearling badge. But I don't know. How to check which package created a user? It is suspected that some packages have scripts run at install time which create user accounts in /etc/passwd. How can I check which package created a specific user?