Prove that there are infinitely many pairs $(k,N)$ of positive integers such that $1+2+\cdots+k = (k+1)+(k+2)+\cdots+N$.   I thought of transforming this into a Pell equation, but I didn't see a way of doing that. Should we show that if the equation is true for some $(k,N)$ it is also true for another?
A positive integer $n$ is called a balancing number if  $$1 + 2 + \cdots + (n − 1) = (n + 1) + (n + 2) + \cdots + (n + r) \tag{1}$$  for some positive integer $r$.     Problem: Find the general solution (closed-form expression) of equation $(1)$, in other words, find $f_1(m)=n, f_2(m)=r$ where $f_1, f_2$ are solution of $(1)$ for any integer $m \in \mathbb Z$.   I have asked a  with a constrain involving Pell's equation.
I know how to use the subfig package to fit many subfigures (each with its own caption) into a larger figure (with a general caption).  I know how to use the sidecap package, the put a figure's caption on the side of the figure.  What I do not know is how to display a caption inside the figure, as if it were a subfigure.  More precisely, I have three subfigures that I would like to fit on a 2x2 layout, with the remaining quadrant used for the caption.    Is that easy to achieve? How can I programmatically pause an NSTimer? I'm using an NSTimer to do some rendering in an OpenGL based iPhone app.  I have a modal dialog box that pops up and requests user input.  While the user is providing input I'd like to "pause" i.e. something like this:  [myNSTimer pause];   I'm using this syntax because I've been doing things like:  [myNSTimer invalidate];   when I want it to stop.  How can I programmatically pause the NSTimer? Why does cancellation of dots $\frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$ work? Re-enabling backspace as "go back" in recent Chrome Since start of this summer, I think that Google removed the option of using backspace for returning to previous page or so.   I was wondering, if there exist any chance, of undo-ing this thing? Is there such thing that would help me, such as downgrading Chrome or using any additional add-ons or something such as this for getting this glorious function back?  It's nearly impossible for me to forget this shortcut (as old habits die hard, lol) and keep pressing it and later on using the mouse to go back. Choice of $q$ in Baby Rudin's Example 1.1 First, my apologies if this has already been asked/answered.  I wasn't able to find this question via search.  My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch 1, Example 1.1 on p. 2.  In the second version of the proof, showing that sets A and B do not have greatest or lowest elements respectively, he presents a seemingly arbitrary assignment of a number $q$ that satisfies equations (3) and (4), plus other conditions needed to show that $q$ is the right number for the proof.  As an exercise, I tried to derive his choice of $q$ so that I may learn more about the problem.  If we write equations (3) as $q = p - (p^2 - 2)x$, we can write (4) as   $$ q^2 - 2 = (p^2 - 2)[1 - 2px + (p^2 - 2)x^2]. $$  Here, we need a rational $x &gt; 0$, chosen such that the expression in $[...]$ is positive.  Using the quadratic formula and the sign of $(p^2 - 2)$, it can be shown that we need   $$ x \in \left(0, \frac{1}{p + \sqrt{2}}\right) \mbox{ for } p \in A, $$  or, for $p \in B$, $x &lt; 1/\left(p + \sqrt{2}\right)$ or $x &gt; 1/\left(p - \sqrt{2}\right)$.  Notice that there are MANY solutions to these equations! The easiest to see, perhaps, is letting $x = 1/(p + n)$ for $n \geq 2$.  Notice that Rudin chooses $n = 2$ for his answer, but it checks out easily for other $n$.  The Question:  Why does Rudin choose $x = 1/(p + 2)$ specifically?  Is it just to make the expressions work out clearly algebraically?  Why doesn't he comment on his particular choice or the nature of the set of solutions that will work for the proof?  Is there a simpler derivation for the number $q$ that I am missing? "IllegalFormatConversionException: d != java.lang.String" when padding number with 0s? How much time have the Straw Hats spent with each other? Tensor product in dual-space I am a bid confused regarding the notation for tensor products when going into dual-space  If $\left| \Psi \rangle \right. = \left| A \rangle \right. \left| B \rangle \right.$ is $ \left. \langle \Psi \right| = \left. \langle A \right| \left. \langle B \right|$ or $\left. \langle B  \right| \left. \langle A \right|$?  My guess is the last choice, since operators should act on   $\left| 0 \rangle \right. = \left( \begin{matrix}   1 \\   0  \end{matrix} \right) $  $\left| 1 \rangle \right. = \left( \begin{matrix}   0 \\   1  \end{matrix} \right) $  so  $\left| 01 \rangle \right. = \left| 0 \rangle \right. \left| 1 \rangle \right. = \left( \begin{matrix}   0 \\   1 \\   0 \\   0  \end{matrix} \right) $  But then, for me, it should be  $\left( \begin{matrix}   0 &amp; 1 &amp; 0 &amp; 0  \end{matrix} \right) = \left \langle 10 \right|$ But this is not correct if i calculate the tensor product, since   $\left \langle 1 \right| \left \langle 0 \right| = \left( \begin{matrix}   0 &amp; 0 &amp; 1 &amp; 0  \end{matrix} \right)$  For instance, shoulden't it be  $\left| 01 \rangle \right. \left \langle 10 \right| = \left( \begin{matrix}   0 &amp; 0 &amp; 0 &amp; 0\\   0 &amp; 1 &amp; 0 &amp; 0\\   0 &amp; 0 &amp; 0 &amp; 0\\   0 &amp; 0 &amp; 0 &amp; 0\\  \end{matrix} \right) $ ?  I hope I have made my confusion clear Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating series etc.  One series that popped into our discussion was $\sum\limits_{n=1}^{\infty} \frac{1}{n^n}$.  Proving the convergence of this series is trivial but finding the value to which converges has defied me so far. Mathematica says this series converges to $\approx 1.29129$.  I tried Googling about this series and found very little information about this series (which is actually surprising since the series looks cool enough to arise in some context).  We were joking that it should have something to do with $\pi,e,\phi,\gamma$ or at the least it must be a transcendental number :-).  My questions are:   What does this series converge to? Does this series arise in any context and are there interesting trivia to be known about this series?   I am actually slightly puzzled that I have not been able to find much about this series on the Internet. (At least my Google search did not yield any interesting results). Zip all files in directory? Energy-momentum tensor transformation C++11 does not deduce type when std::function or lambda functions are involved Why is the Star Trek 2009 reboot called a reboot? Doctor Who (2005) isn't called a reboot. It is called a continuation of the old series. This was made possible by that regeneration ju-ju and stuff.  On the other hand, Star Trek 2009 reincarnation is often called a reboot. Nobody dared call it continuation. I don't think I need to do citation because it's common.  Now, if we actually go into the details:   The new universe contains two Spock. One is from the original timeline. The Spock Prime even gives data from the original timeline like trans-warp equation, Khan etc. We see other past data from before the timeline diverged. For example, Scott told that he transwarped Admiral Archer's dog. "Archer's dog" existed in Star Trek: Enterprise TV series.   So, why didn't the legacy survive in-universe? Why isn't it considered a continuation of the old Star Trek TV shows and movies?  Or, is the reboot word used by mistaken fans and media only?  Update: After the release of Star Trek: Beyond (2016), this is no longer a confusion as the movie showed a photo from original timeline. MySQL: Allow both remote AND local connections Suppose I have a paper with the following text:     Roses are red, violets are blue, sugar is sweet, and so are you [3],[4].   As one can see, the author got the information from two other references. If I want to add this information in my thesis, do I also have to reference [3] and [4] or can I just reference this particular paper, I got the information from? Let $G(r, n)$ be the Grassmannian of the set of all $r$-planes in a $n$-dim vector space. How to show that the cohomology of a Grassmannian has a basis consisting of the equivalent classes represented by Schubert cycles? I am confused since I don't know how to compute the cohomology of a variety (how to construct the co-chain complex). I am really appreciate if you can compute the cohomology of, for example, $G(2,4)$ and show that it has a basis consisting of the equivalent classes represented by Schubert cycles. How to prove floor identities? Why do I keep getting told I posted less than 15 characters when I posted more than 15? I tried to post the following comment and it claims it was less than 15 when it was more.  Based on Tim's comment and testing I believe it would not let me post because with @Shadow removed it is less than 15. However, the characters left message indicates I have passed the 15 character minimum thus it is wrong.  This is the bug I'm reporting:  The characters remaining message and the characters left message do not take into account striping out @name prefixes. I have 2-3 different column names that I want to look up in the entire DB and list out all tables which have those columns. Any easy script? Multiplication in Permutation Groups Written in Cyclic Notation