Why is my shadow deformed and not the shadow of the ceiling?
Why do shadows from the sun join each other when near enough?
Drop or change automatic migration-rejection based on non-existing tags I believe this feature creates more issues than it solves and thus would like to see it gone. (This is an out-growth of , which I repost here as it concerns several/all sites.)  The main reason why I think so is that there can be numerous reasons other than the subject being off-topic why some tag does not exist on some site, for one thing it could be a spelling, typesetting, or other formalistic reason. For example, for the two sites I know best MathOverflow and Mathematics hardly any of the most popular tags have a strictly matching tag on Mathematics but only because on MO there is the convention that certain tags are formatted in a specific way, which is not in place on Mathematics (for example nt.number-theory vs number-theory).  On top of that a user choosing the wrong site in the first place is likely not too well aware of the tags of the site they are posting too so that the question might be poorly tagged in the first place.         There is an additional detail I do not understand (it is obsolete if the actual request should be followed but in case it is not):   Even if on the source site a tag has a synonym and that synonym exists as a tag on the target site the migration is rejected. I do not understand the logic behind this at all. Possibly this is just an oversight.   I could also think of some other ways to improve the process while keeping the idea, but since I believe that it is best dropped I do not (yet) elaborate on this. Find the exact value of the infinite sum $\sum_{n=1}^\infty \big\{\mathrm{e}-\big(1+\frac1n\big)^{n}\big\}$ How can we find the exact value of the infinite sum  $$ \displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\Big)^n\right\}? $$  This problem appears in:  , p.114. Possible Duplicate:          I want to know some example of a continuous function which is continuous at exactly one point. We know that $f(x)=\frac{1}{x}$ is continuous everywhere except at $x=0$. But i think this in reverse manner but i dont get any example. So please help me out! What are the recommended textbooks for introductory calculus? I've already taken my calculus sequence and I'm interested in brushing up and staying sharp on the basics. So far, my calculus background is limited to single-variable calculus, which I applied in my physics sequence and engineering statistics. I learned a little bit of vector math in physics as well, but I'm not strong in it. My mathematical strengths lie in the discrete mathematics.  I'm particularly interested in studying for my IEEE Certified Software Development Associate exam - 10% of the exam is mathematics (calculus, differential equations, and statistics). Calculus is one of the mathematics knowledge areas that a software engineer should be competent in, so I'm looking for not only solidifying the courses that I've already taken, but moving forward with slightly more advanced topics, at an undergraduate mathematics level.  I'm currently using the textbook from my calculus sequence, but I'm interested to know what books other people used to learn calculus and if the book is potentially worth checking out.  What books are available to help me? How are local variables referenced in closures? The possessive 's in time expressions Which is the correct answer?         I'll be there in a _____ time.       A) day or two   B) day or two's   C) day's or two's Z-axis issue with the floating review info panel, permits various elements to flow over it Does RSA work for any message M? I decided to read the original RSA paper A Method for Obtaining Digital Signatures and Public-Key Cryptosystem because of a question I had about RSA (which is not the question I'm about to ask, but may be a question on here at some time) and came across something I had never seen before in any textbook description of RSA (Note: this comes from the beginning of ).  for any integer (message) $M$ which is relatively prime to $n$, $M^{\phi(n)} \equiv 1\pmod{n}$  The part I don't recall ever hearing in class, or coming across in a text book, or seeing in any description of RSA I've ever read is that $M$ and $n$ must be relatively prime. So, I decided to play around with a toy example of RSA to see what would happen. I use $p=13$, $q=31$, $e=7$ and $M=2p=26$ (for completeness: $d=103$, $\phi(n)=360$, $n=403$). I observed the following using my toy instance of RSA: $M^e\equiv M\pmod{n}$, $M^e\equiv 0\pmod{p}$ and $M^e\equiv M\pmod{q}$. Do these equivalences hold for all $M,p,q$ when $M$ and $n$ are not relatively prime? How would I prove this? What is the real-world effect of this? I have  with a bounty that will expire in a few days. It looks like nobody will get the bounty; only one answer was posted during the bounty period, and it has no upvotes.  In cases like this, the bounty is thrown away.  Why isn't it refunded instead, if no answerer has earned it? See screenshot:    Curiously, there are older badge achievement messages further down the list which display properly. Solving 4 Quadratic Simultaneous Equations Consider the equations:   $$\begin{cases} ab+c+d=3\\ bc+d+a=5\\ cd+a+b=2\\ da+c+b=6 \end{cases}$$   Where $a,b,c,d \in \mathbb{R}$. How can we find $a,b,c,d$?   The furthest I've got is by adding the first two equations and the last two equations together to get: $ab+bc+c+d+d+a=(b+1)(a+c)+2d=8$ and $(d+1)(a+c)+2b=8$ We can rearrange these (assuming $b,d \neq -1$) to get $$\frac{8-2d}{b+1}=\frac{8-2b}{d+1} \implies -d^{2}+3d+4=-b^{2}+3b+4$$ Which leads us further to $$b^{2}-d^{2}-3(b-d)=0 \implies (b-d)(b+d-3)=0$$ Now since $b=d$ gives us an absurdity (equations 1 and 4 reduce to 3=6), we must have $b+d=3$. This is as far as I can get unfortunately, since trying the same approach with the other pairs doesn't work in the same way.  Help would be much appreciated; the problem is from the BMO1 2004 paper. Interactive / Real time code compilation for stackoverflow.com. That will improve the answer correctness of the stackoverflow.com.  When adding a new code block, let the user try to compile the code and see the result. It will be useful to demonstrate class designs, algorithms, and logics. House wiring and grounding. How do you retrofit an old house and ground it? A contractor looked at our house and told me that you can't fix a house (with old wiring) that has some outlets that are not grounded.  He said that you can go on YouTube and search the internet and it will tell you that you can.  He says that the only way to properly fix and ground everything is to do a whole house rewire.  Is he correct?  If, theoretically, you have an old house that has no ground whatsoever, can you run a grounding wire from the panel to all of the outlets to fix this?  updated  We have BX/AC wiring. And also apparently we have "...old-style AC (BX) with cloth insulated wires in a paper overall wrap under the spiral armor (no bonding tape)" Courtesy of @ThreePhaseEel. How to escape @ characters in Subversion managed file names? Date Format In RUBY When I login to my ubuntu system, it show me this this message, How can I see it any time? I tried to run . .bashrc but it didn't worked.   Welcome to Ubuntu 12.10 LTS (GNU/Linux 3.2.0-24-virtual x86_64)   * Documentation:  https://help.ubuntu.com/    System information as of Sat Jul 13 06:16:35 UTC 2013    System load:  0.55              Processes:           96   Usage of /:   8.1% of 68.74GB   Users logged in:     0   Memory usage: 55%               IP address for eth0: 198.xx.xxx.xxx   Swap usage:   0%                IP address for eth1: xxx.1x0.xx.xxx    =&gt; There is 1 zombie process.    Graph this data and manage this system at https://landscape.canonical.com/  223 packages can be updated. 134 updates are security updates.  Last login: Mon Jul  8 17:47:05 2013 How can I display keyboard shortcuts as I press them? supervisor field on the page of the title with Beamer I am editing a presentation of a research project, I am wondering if there is a field for "supervisor" on the page of the title, for instance \supervisor(which actually does not work)... If not, where do people conventionally add it? I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark system. Namely:  Let the tetraquark system be:           $QQ\bar q\bar{q}$  So, those $Q$ heavy quarks interact and can be understood in the $3\bigotimes 3$ representations for $SU(3)$, and $\bar q\bar q$ can be understood as the $\bar 3\bigotimes\bar 3=3\bigoplus\bar 6$ representations for $SU(3)$.  The interaction diagram for the $Q$ heavy quarks is shown in the following scheme, which is an analogy to the Feynman Diagram where a gluon is understood to be exchanged between upper and lower arrows:  $i'\xrightarrow{(T_{a})^{i'}_{i}}i ,  {}^iQ$  $j'\xrightarrow{(T_{a})^{j'}_{j}}j ,  {}^jQ$  So, as usuall we consider the elements of $3$ representation as $u^i$, and therefore the elements of $\bar 3$ representation as $v_{j}$.  I must understand how does the following expression transform:  $(T_{a})^{i'}_{i}(T_{a})^{j'}_{j}\cdot(1/2)\cdot (w^iv^j+w^jv^i)=\Xi\cdot(1/2)\cdot(w^{i'}v^{j'}+w^{j'}v^{i'})$  In other words, how can one give the correct value of $\Xi$?  Note1: $T_a$ is understood to be the $a-ith$ generator of the associated Lie Algebra, which come from the definition of the Gell-Mann Matrices.  $(T_{a})^{i}_{j}\propto(\lambda)_{ij}$, in matrix notation.  Note2: I understand the Young Tableaux and they are not intended to be used here, so please don't try to explain this situation through the tables.   Note3: I assume this is a purely solvable mathematical problem and doesn't need a deep insight into the physical concepts...  Also requesting information about how to understand the tensor products in these cases, since I have been checking many sources but the definitions seem not clear enough. When I try to create a reaction from chemmacros/reactions I got the following error. I dunno what I am missing. Please help. Thank you.     line 11: Undefined control sequence. \end   line 11: Missing number, treated as zero. \end   line 11: Missing number, treated as zero. \end     Minimum Working Example    \documentclass[12pt,a4paper]{article} \usepackage{chemmacros} \chemsetup{modules={reactions}} \begin{document} \begin{reaction}     A + B -&gt; C \end{reaction} \end{document}