Sometimes when I walk around the world, small red triangle appears on the mini map, next to some other marker (e.g. plant or building) then disappears when I keep walking:        What do these triangles mean?
I'm actually at the cemetery at White Orchard. There is a white question mark and some little red arrows.  What are this little red arrows?  Do they point to something dangerous?
Is the collapse of the wave function inherently time asymmetric? What does different focal length offer me? At what point is the ~/.bashrc file created? What are the benefits of playing an alignment other than True Neutral? Why are there greyed / dimmed / faded questions now? I just refreshed the [iphone]-tag-site, and I'm seeing all kinds of questions with what appears to be a grey overlay (see image). What is this for/why am I seeing this? I have never seen anything like this before.. How do I check an iOS device's battery health? How do I with ufw deny all outgoing ports excepting the ones I need? How do you mass-delete emails in iOS Mail? I don't know if this is documented anywhere, but just to bring it to current attention… Why do light sources appear as stars sometimes? See for example this photo:    From my experience, the longer the exposure the stronger can this effect be observed. Is this correct? Are there any other factors that influence the creation of these stars (is there a better word for it, by the way?) And what exactly does technically happen? I'm trying to prove that the converse of Lagrange's theorem is true for finite abelian groups (i.e. "given an abelian group $G$ of order $m$, for all positive divisors $n$ of $m$, $G$ has a subgroup of order $n$"). This is an exercise from a book, and it is in the section on finite abelian groups, so I know I have to use the fundamental theorem of finite abelian groups. I have come up with a proof, but it seems a bit messy, and I'm not entirely sure if it's correct. It is given below.     Let the order of $G$ be $m$ = $p_1^{\alpha_1} \ldots p_k^{\alpha_k}$. It is known that $G$ is a direct product of $p$-groups, say:      $$G = G_1 \times \ldots \times G_k$$      where each $G_i$ is a $p_i$-group. By the fundamental theorem of finite abelian groups, each $G_i$ is isomorphic to a direct product of cyclic groups of the form      $$\mathbb{Z}_{{p_i}^{\beta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\beta_l}},$$      where $\beta_1, \ldots, \beta_l$ are positive integers such that $\sum_{j=1}^l \beta_j = \alpha_i$. Now if $n$ divides $m$, then we must have      $$n = p_1^{\gamma_1} \ldots p_k^{\gamma_k}$$      for some $\gamma_1, \ldots, \gamma_k$ with $0 \leq \gamma_i \leq \alpha_i$.      Claim: Each $G_i$ has a subgroup of order $p_i^{\gamma_i}$      Proof: As above, we have that       $$ G_i \cong \mathbb{Z}_{{p_i}^{\beta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\beta_l}} $$      where $\beta_1, \ldots, \beta_l$ are positive integers such that $\sum_{j=1}^l \beta_j = \alpha_i$.      Now since $0 \leq \gamma_i \leq \alpha_i$, we can find   $l$ numbers $\delta_1, \ldots , \delta_l$ such that $\gamma_i = \sum_{j=1}^l \delta_j$, and $0 \leq \delta_j \leq \beta_j$. (This choice of numbers is not necessarily unique).      Then $p_i^{\delta_j} | p_i^{\beta_j}$ for each $j = 1, \ldots , l$. Hence, for each factor $\mathbb{Z}_{{p_i}^{\beta_j}}$, there exists a subgroup of order $p_i^{\delta_j}$, namely $\mathbb{Z}_{{p_i}^{\delta_j}}$ (using the fact that the converse of Lagrange's theorem is true for finite cyclic groups). Taking the direct product of each of these subgroups, we get a new subgroup $G_i'$ of $G_i$:      $$G_i' \cong \mathbb{Z}_{{p_i}^{\delta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\delta_l}}$$      The order of this subgroup is $p_i^{\delta_1} \times \ldots \times p_i^{\delta_l} = p_i^{\delta_1 + \ldots + \delta_l} = p_i^{\gamma_i} $. So we have found a subgroup of $G_i$ of order $p_i^{\gamma_i}$, as required.      So each factor $G_i$ in the product $G = G_1 \times \ldots \times G_k$ has a subgroup $G_i'$ of order $p_i^{\gamma_i}$.       Therefore, $G$ has a subgroup   $$G_1' \times G_2' \times \ldots \times G_k'$$       of order $p_1^{\gamma_i}...p_k^{\gamma_k} = n$, which completes the proof.   I have two questions about this: firstly, does this proof seem to work? Secondly, is there a way to make the proof more concise (e.g. a way to prove the statement without using all these indices)? How can I use an array modifier to create individually manipulatable objects? I often use an array modifier (or two) to quickly create a grid of identical or symmetric objects, which can be really useful when modeling all sorts of man-made objects. Once I've made the grid, though, is there any way for me to make it so that each of the objects are individually manipulatable?  Simply applying the modifier does not seem to be the full answer, since it still only allows me to select the grid-of-objects as a whole in object mode. Is there any way to separate them so I work with them each individually? Is there a list of error codes for Minecraft? Similar to  on the bugs in Minecraft, I would like a complete list of what error codes exist in Minecraft and how they occur, as well as how they can be resolved. Parse HTML in Android $2 \otimes_{R} 2 + x \otimes_{R} x$ is not a simple tensor in $I \otimes_R I$ I am a new user for Ubuntu. I downloaded 13.10 yesterday and made a bootable USB with universal USB installer on my Lenovo IdeaPad Z580 laptop which is now using UEFI; everything appeared OK.  When I booted from the LiveUSB I got the choices of trying or installing Ubuntu but both of them keep giving me these error:  [ 1.929082] kernel panic-not syncing vfs unable to mount root fs on unknown-block (0.0)   Please tell me what is going wrong? What should children call a step-grandparent? My mom remarried to a great guy who has been absolutely wonderful to my kids. Ever since my mom and he were dating my kids have called him Mr. Joe. Lately my mom has been bugging me to get the kids to call him Grampa.  I agree that he deserves more respect than Mr.  The problem is that he isn't their real grampa so I don't really want them calling him that. Does anybody have any suggestions on what we should have the kids call him? They are 9 and 12 years old.  EDIT: I appreciate the comments regarding different variations of grampa and hearing how others have no issues with calling their step-granddad grampa. But I was hoping maybe someone has some suggestions for names that show the importance of the step-granddad without calling him some variation of the grampa name but still has a feeling of warmth and family in the name (I hope that makes sense). Here comes Bob.  Bob is coming here.  Would anyone please explain what is  the difference between these? In $\mathbb{R}$ with standard topology the closure of the above set $M$ is $M \cup 0$. I fail showing this by purely topological means. I.e. not using the fact that this holds in the reals as a metric space and deduce it for the induced topology.  I assumed that there exist a closed subset $A$ containing $M$ but not 0. Hence, the complement of $A$ is an open subset containing 0. Does this already yield a contradiction or do I need a different approach?  Could you help me on this? In the discussion from a question I asked several days ago (), someone suggested that I use general.runalg() instead of processing.runalg() for running QGIS algorithms.    I've now used both, and haven't noticed a difference in either performance or syntax between them.  I'm just wondering what sort of differences there are between these two methods, and which one I should be using. How to check if installing a package will require a reboot?