Lie algebra of 2 by 2 matrces that are upper triangular and their characteristics that define them
Lie algebra of $\left(\begin{smallmatrix}a & b\\ & a^2\end{smallmatrix}\right )$ in $GL_2(\mathbb{R})$
How to install (old) packages for Ubuntu 9.04? Based on some excellent feedback by Mark here (), today I was able to use the vmware converter to clone my production server for a staging server.  However the nic won't come up no matter what I do. I attempted to inistall vmware tools, as I suspect that the fact that it is not installed might prevent the nic from working. (I have the nic set as a vmxnet3 card in the vm settings). The install failed because there were several dependencies missing as well as the Linux headers. Given that Ubuntu 9.04 has been EOL'd, the packages I need to install to get the vmware tools to install are no longer available. I doubt the ubuntu 9.04 install CD has the packages on it.   What are my options? I'd rather not upgrade the version of Ubuntu yet, as the point of the vm right now is to maintain parity with the production server. Might I have better luck resetting the driver to use vmxnet2 instead of the vmxnet3?  Thanks in advance!  Chris Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction I recently proved that  $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$  using mathematical induction.  I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any other proofs. Merge two unadjacent partitions I have three partitions on my drive, arranged something like the following:   ___________ | 1 |  2 | 3| -------------   I wish to merge partitions 1 and 3.  Is there a way to do this, or do I have to manually "slide" partition 2 over to partition 3, and then merge the newly open space where partition 2 used to be?  Note: I do not have the space to copy partition 2 to partition 3 (2 is 200GB and 3 is 65GB). I gave a quick answer to part of  which had not been covered by previous answers, trying to clarify the reason you would say time of decoding but not decoding’s time. I said it was ’s usually indicates possession, but of course there were several counterexamples that would have occurred to me after a moment’s consideration, and these where helpfully supplied:   Britain’s climate two days’ time  a day’s work  the sun’s rays   I am still of a mind to say that possession of some sort is what allows the ’s. Even though the sun does not have title to its rays, they do belong to the sun. Now, at the risk of duplicating the original question and/or being pigheaded, I am curious as to why time of decoding but not decoding’s time is correct, if not for the reason I gave. If $f$ is entire and $|f|\geq 1$, then show $f$ is constant. I know I'm going to use Liouville's Theorem, but my main question is why is $1/|f(z)|$ entire as well if $f$ is entire? Is this just a basic property: if $f$ is entire, then $1/f$ is entire? Thanks for the help. Is there an exception to the prohibition against ending a sentence with “ ’s ” at work here? The ’s can be used as a contraction representing a weak, unstressed word that is not pronounced.  It allegedly cannot occur in sentence final position.        She is not ready, but he is.   She’s not ready, but he is.   She’s not ready, but *he’s.      The last one is not grammatical there.  Similarly, here the last one is not grammatical:        He has not started yet, but she has.   He’s not started yet, but she has.   He’s not started yet, but *she’s.       claims this is because you cannot end a sentence with a weak form.  It backs up its point with reasonably scholarly documentation.  However, this does not seem to be invariably true.        Shall we go to the movies?  Yes, let us go.   Shall we go to the movies?  Yes, let’s go.   Shall we go to the movies?  Yes, let’s.      Unlike the third example in the first two sets, here in this case the third example is indeed grammatical, despite ending with a weak form.  Is this an exception to the rule, or is there another rule at work here? Moderncv adding date of birth to personal information I'm doing my cv with the moderncv package and I'm using the classic style. I would like to add a date of birth line directly above the adress. Does anyone know how to do this? I've tried various solutions but they all seem to be broken. Why Did Anakin Skywalker Believe that Turning to the Dark Side of the Force Would Save Padmé? Just as the question asks: Why did Anakin come to believe that Padmé could be saved if he embraced the Dark side of the Force? Padmé clearly didn't believe it. Why was Anakin so driven to save Padmé through the dark side? Intuition on the Kullback–Leibler (KL) Divergence How to prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$ Ubuntu 18.04 No WiFi adapter found [Dell Inspiron 3542.] Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$ Recently, I ran across a product that seems interesting.  Does anyone know how to get to the closed form:  $$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$  I tried using the identity $\cos(x)=\frac{\sin(2x)}{2\sin(x)}$ in order to make it "telescope" in some fashion, but to no avail. But, then again, I may very well have overlooked something.  This gives the correct solution if $n$ is odd, but of course evaluates to $0$ if $n$ is even.   So, I tried taking that into account, but must have approached it wrong.   How can this be shown? Thanks everyone. What method in the String class returns only the first N characters? Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$ Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. I now want to prove that for every $x&gt;0$ $$ P\left( \sup_{t \geq 0 } M_t &gt; x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{M_0}{x}. $$ My approach: I thought that rewriting the conditional probability to an expectation would help so we obtain that we must prove: $$ \mathbb{E} \left[ 1_{\sup_{t \geq 0 } M_t &gt; x}  \mid \mathcal{F}_0 \right] = 1 \wedge \frac{M_0}{x}.$$ A hint to me was given that I should consider stopping the process when it gets above $x$. Thus a stopping time that would do this is $\tau = \inf\{t\geq 0 : M_t&gt;x\}$. But now I'm stuck as I want to apply optional sampling results but we have an indicator which complicates things. How could I proceed from this? Any help is appreciated! Assign VNC password using script I am using a script to automatically set up a computer. I need to assign a password to the vnc server for the user, which is normally done using the vncserver command. However, it prompts for the user to enter and re-enter their password, neither of which the script is capable of doing. So, how can I set up the VNC password without an interactive prompt?   Thanks. What are the options for traveling to the airports in Paris? Paris has 3 airports (Orly, Roissy, Beauvais), what are the options to go to or to come from there?  (There is also Vatry (XCR) but there is very few flights and i see the shuttle system is currently off.) Pressure variation in a sound wave Single word to denote date and time I want one word that means Date and Time. For example in "the date and time of the event will be announced later" I want to use single word instead of "date and time" Why is this sentence wrong? "He went to home." I have  a  requirement where a future method need to be called from batch apex ,I came across various articles stating that a future method could not be called from the batch. Is there any alternate solutions for that?