Does my age depend on the current timezone?
Celebrating the same new year several times over?
What is the ideal data type to use when storing latitude / longitude in a MySQL database? Bearing in mind that I'll be performing calculations on lat / long pairs, what datatype is best suited for use with a MySQL database? As far as I can tell, it's something like this:  damage reduction = armor / (armor + 100)   Is this accurate? Could Hermione really free the Hogwarts house-elves with her knitted hats? What are good books/other readings for elementary set theory? I am looking to expand my knowledge on set theory (which is pretty poor right now -- basic understanding of sets, power sets, and different (infinite) cardinalities). Are there any books that come to your mind that would be useful for an undergrad math student who hasn't taken a set theory course yet?   Thanks a lot for your suggestions! Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist? Does $1.0000000000\cdots 1$ (with  an infinite number of  $0$ in it) exist? Which email was the verification email sent to? I started my app today and it said that I needed to verify my email. But I don't know the email I chose. I logged on every email in but there was never a verify email. Is there a way to know your email on Pokemon Go? dd aplied to dev USB before connect corrupt /dev/sdX and dont work I have a USB where I'm coping an ISO of 5 Mb aprox with a small OS with its own grub. If I connect the device and check it sudo fdisk -l I get:  ... Device     Boot Start   End Sectors  Size Id Type /dev/sdb1  *        1  9551    9551  4.7M cd unknown   If I copy with dd the ISO sudo dd if=my_os.iso of=/dev/sdb   14852+0 records in 14852+0 records out 7604224 bytes (7.6 MB) copied, 0.538487 s, 14.1 MB/s   It works perfectly. An boot from BIOS in other PC.  The problem is, sometime I forget to connect the USB after update the BIOS, so I do ff WITHOUT the USB connected,  the output is:  14852+0 records in 14852+0 records out 7604224 bytes (7.6 MB) copied, 0.00987684 s, 770 MB/s   Now, the /dev/sdb register is corrupted, the copy speed (770 MB/s) have no sense, and cannot user again dd until reboot.  I have tried to repair it with ddrescue and it doesn't work  sudo ddrescue /dev/zero /dev/sdb conv=noerror,sync   My question is, what is happening exactly and if is possible, how to repair /dev/sdb to avoid reboot and dd start working again?  Thanks. PDF Editing in PHP? Does anyone know of a good method for editing PDFs in PHP? Preferably open-source/zero-license cost methods. :)  I am thinking along the lines of opening a PDF file, replacing text in the PDF and then writing out the modified version of the PDF?  I have programmatically created PDF files in the past using FPDF, but found it a little unwieldy at times. I'm studying abstract algebra for the first time.  let $F$ be a field. I learned that polynomials over field $F$ means polynomials with coefficients in $F$ (denoted by $F[x]$). But when I keep studying, I get to confused the word, 'polynomial functions from $F$ to $F$'.  Firstly, I consider this as induced functions from $F[x]$, so I guess all coefficients of these functions are in $F$. But let $F_{4}$ be a finite field consisting of four elements $\{0, 1, a, b\}$. and let $f: \{0,1\} \rightarrow \{0,1\}$ be a function defined by $$ f(x)=ax^2-ax$$ Since $\{0,1\}$ is a subfield of $F_{4}$ and $f(0)=0, f(1)=0$, I think $f$ is a example polynomial functions from $F$ to $F$ of which coefficients are not in $F$. Am I wrong? if so, please help me.  I encounter this problem while solving 'the set of all polynomial functions from $\Bbb{Z_{3}}$ to $\Bbb{Z_{3}}$ cannot possibly be isomorphic to $\Bbb{Z_{3}}[x]$'. Group theory proof of Euler's theorem ($a^{\phi(m)} \equiv 1\mbox{ }(\mbox{mod }m)$ if $\gcd(a,m)=1$) From A Classical Introduction to Modern Number Theory by Ireland and Rosen, page 33:     Corollary 1 (Euler's Theorem). If $(a,m) = 1$, then $a^{\phi(m)} \equiv 1\,(m)$.      Proof. The units in $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. If $(a,m) = 1$, $\bar{a}$ is a unit. Thus $\bar{a}^{\phi(m)} = \bar{1}$ or $a^{\phi(m)} \equiv 1\,(m)$.   If I'm interpreting this correctly, this proof implicitly uses the fact(?) that if $G$ is a group and $x\in G$, then $x^{|G|}=1$, where $1$ is the identity element of $G$.  If this is indeed true, can someone explain why? (I have had no prior exposure to group theory.) "approach to address" or "approach to addressing"? Depending on our point of view, we offer different approaches to address(ing?) a same issue.   Should I use &mdash;ing in this context, and if yes, why? From the man page on git-merge, there are a number of merge strategies you can use.     resolve -  This can only resolve two heads (i.e. the current branch and another branch you pulled from) using 3-way merge algorithm. It tries to carefully detect criss-cross merge ambiguities and is considered generally safe and fast. recursive -  This can only resolve two heads using 3-way merge algorithm. When there are more than one common ancestors that can be used for 3-way merge, it creates a merged tree of the common ancestors and uses that as the reference tree for the 3-way merge. This has been reported to result in fewer merge conflicts without causing mis-merges by tests done on actual merge commits taken from Linux 2.6 kernel development history. Additionally this can detect and handle merges involving renames. This is the default merge strategy when pulling or merging one branch. octopus -  This resolves more than two-head case, but refuses to do complex merge that needs manual resolution. It is primarily meant to be used for bundling topic branch heads together. This is the default merge strategy when pulling or merging more than one branches. ours -  This resolves any number of heads, but the result of the merge is always the current branch head. It is meant to be used to supersede old development history of side branches. subtree -  This is a modified recursive strategy. When merging trees A and B, if B corresponds to a subtree of A, B is first adjusted to match the tree structure of A, instead of reading the trees at the same level. This adjustment is also done to the common ancestor tree.   When should I specify something different than the default?  What scenarios are each best for? How to proof that $\forall n \ge 1: \sum_{i=1}^n (i(i+1))^{-1} = n(n+1)^{-1}$ using mathematical induction So for a better presentation:   I need to prove this using mathematical induction, so i started with a base of $n=1$  Since $i$ starts at $i = 1$ I got this:  $$\frac{1}{i(i+1)} = \frac{n}{n+1}$$ $$\frac{1}{2} = \frac{1}{2}$$  So since this was true I went to the induction part: I chose $n = k$ and assumed this was true:  $$\frac{1}{i(i+1)} = \frac{n}{n+1}$$ $$\frac{1}{i(i+1)} = \frac{k}{k+1}$$  but I got stuck here since I didn't know if I needed to increment $i$ or not and because I didn't know if I needed to add the summation symbol or not and needed to do a few things with it. Whenever I start an application like Terminal or Preview, they are always opened as a small window on the top left corner of the screen. Is there any way to set them to open in the center of the screen? Preferably maximized too? 2D and 3D vectors in Tikz I would like to draw some 2D and 3D vectors in Tikz with certain properties. The following primitive graph sums what I desire:   I just need the first quadrant of the Cartesian coordinate system for my representation.  I need the vector to be connected to the respective values on the distinct dimensions(not a projection, just orthogonal dashed lines).   The code segment below does what I want to achieve in 3D with two exceptions:  The projections on the xy plane and z dimension are given instead of connecting the head of the vector to the corresponding value in each dimension.  Numerical values are missing in dimensions.   Code: \documentclass[tikz]{standalone} \usepackage{tikz-3dplot} \begin{document}  \tdplotsetmaincoords{60}{120} \begin{tikzpicture}     [scale=3,     tdplot_main_coords,     axis/.style={-&gt;,blue,thick},     vector/.style={-stealth,red,very thick},     vector guide/.style={dashed,red,thick}]  %standard tikz coordinate definition using x, y, z coords \coordinate (O) at (0,0,0);  %tikz-3dplot coordinate definition using r, theta, phi coords \tdplotsetcoord{P}{.8}{55}{60}  %draw axes \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};     \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};     \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};  %draw a vector from O to P \draw[vector] (O) -- (P);  %draw guide lines to components \draw[vector guide] (O) -- (Pxy); \draw[vector guide] (Pxy) -- (P); \end{tikzpicture} \end{document} Confused about AMD/Intel When I go to the downloads page to get Ubuntu, it automatically downloads the AMD version, but my computer has no hardware with AMD on it. I only have Intel and Nvidia. Also, when I install this AMD version on my computer, it works (other than the constant random freezing of the computer when I use Ubuntu). I don't really understand why the amd version works on a computer with an intel processor. How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$? I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded into the Euclidean space of the same dimension. So I wonder if all manifolds can be embedded into a same dimensional Euclidean space when we cut out a measure zero set? Can anyone prove it or disprove it by giving me some counterexamples? SSL_read failing with SSL_ERROR_SYSCALL error How to uninstall a package installed from snapcraft? I used this command line  sudo snap install pwgen-tyhicks   to install pwgen but I can't findout how to use it. So I want to uninstall it. How to  uninstall it? halt, poweroff, shutdown, reboot don't require root? For some reason, I can run halt, poweroff, shutdown, and reboot as a regular user without root. How can I fix this? Is it something to do with user groups, or is it a seting somewhere?