How can I find all the combinations of elements in a set of arrays?
Get the cartesian product of a series of lists?
Center figure that is wider than \textwidth ON an American street, but IN a British one. Do the twain ever meet? In the United States, we say that someone lives on a street, whereas I've noticed that British people say in. For instance:     Bubba lives on Washington Street.   Colin lives in Cavendish Avenue.   I believe we both would use at when a number is given. For instance:     Bubba lives at 16 Washington Street.   Colin lives at 7 Cavendish Avenue.   I don't think it matters if it's a road, avenue, street, circle, or lane — as far as I know, in America, we always live on a street. Does it vary in British English, or is it always in? If it varies, what are the rules? What is `^M` and how do I get rid of it? Function is Continuous When Restricted to Closed Sets Which Cover the Space Suppose that $\{A_i\}$ is a finite collection of closed sets in $X$ such that $X = \bigcup A_i$, and $f : X \to Y$ restricted to each $A_i$ is continuous. Then $f$ is continuous.   Okay. The base case ($n=2$) is just the pasting lemma. I am having a little trouble with the inductive case, specifically with what the hypothesis is and how to deal with continuity with respect to subspaces. I believe the inductive hypothesis is to assume that any function over a topological space that is the union of $n-1$ closed sets such that the function restricted to each of these $n-1$ closed set is continuous must be continuous over the whole space. Then I assume I have a topological space $X = \bigcup_{i=1}^n A_i$, where each $A_i$ is closed, and a function $f : X \to Y$ such that $f|_{A_i}$ is continuous for each $i$. Thus $X = \bigcup_{i=1}^{n-1} A_i \cup A_n$, where $C = \bigcup_{i=1}^{n-1} A_i$ is closed since it is a finite union of closed sets. If I can show $f|_C$ is continuous, then I simply apply the pasting lemma to finish the problem. Give $C$ the subspace topology, and notice that each $A_i$ is closed in $C$. Thus we have a topological space $C$ that is the union of $n-1$ closed. But do I still have continuity $f$ restricted to $A_i$ since I am no longer considering them subspaces of $X$ but of $C$? When one says $f|_{A_i}$ is continuous, is one saying that $f$ is continuous when restricted to $A_i$ and $A_i$ is given the subspace topology?  EDIT:  I think I figured out the lemma I need. Claim: Let $A$ and $C$ be subspaces of $X$ such that $A \subseteq C$. Topologize $C$ with subspace topology with respect to $X$. Then the subspace topology on $A$ as a subspace of $X$ (call it $\tau_X$) is the same as the subspace topology of $A$ as a subspace of $C$ (call it $\tau_C$).   Proof: Let $U$ be open in $X$. Then $A \cap U$ is some open set in $A$ as a subspace of $X$. Since $C$ is a subspace of $X$, $C \cap U$ is open in $C$ as a subspace of $X$. Then $A \cap (C \cap U) = (A \cap C) \cap U = A \cap U$ is open in $A$ as a subspace of $C$, thereby proving $\tau_X \subseteq \tau_C$. Now let $O$ be some open set in $C$ as a subspace of $X$. Then $A \cap O$ is some open subset of $A$ as a subspace of $C$. But $O = C \cap U$ for some open set in $X$. Therefore $A \cap O = A \cap (C \cap U)= A \cap U$, which is also open in $A$ as a subspace of $X$. Hence $\tau_C \subseteq \tau_X$  Kind of obvious. But it was hard to spot the need for this lemma when I was considering subspace of subspaces and continuous functions, etc. Am I right in thinking this is the lemma I need? What's the difference between correlation and simple linear regression? In particular, I am referring to the Pearson product-moment correlation coefficient. Subtracting (erasing) polygons from polygon using ArcGIS for Desktop? I am compiling LaTeX files with Texmaker on Ubuntu 12.04 system.   When I run pdflatex and then view the pdf file (internal), it does not show me the exact place where I am editing on Texmaker i.e. the generated pdf file does not show me the position of the cursor.   But some days before it was working fine. I don't know what causes this problem.  I already tried uninstalling and then reinstalling, I did update and upgrade too. But the problem still persists. How do I straighten UV maps? How can I work with customize completion? I want to add some new auto-completion but I don't know How the wizard works. It's here: User>Customize Completion I don't know what should be writen in the box. How to prevent a scrollview from scrolling to a webview after data is loaded? How to read exponential expressions, e.g., "2^16"? How do you say the mathematical function in English:     x^y (or xy)   For example, how do you say      2^16  (or 216)   I know ^ means 'power' or 'exponentiate', but that is the name of the operation, not how you pronounce its use.  If y = 2, it is 'x squared', y=3 then 'x cubed'. What if y is some other arbitrary number? How can there be animals on the surface in Zion's world? It is apparently canon that  in Neo's day.  However, in the first movie, The Matrix, Morpheus tells Neo that one of the things known for certain is that humans "burned the sky" in order to deny solar engery to the machines, which in turn led to the humans who survived the War being plugged into the Matrix to provide bioelectric energy to supply the machines.  If there's not enough sunlight for solar power, how can there be enough to grow plants to serve as food for animals?  Low levels I can understand -- house plants, at least, thrive in conditions too dim for solar panels to produce cost-effective power -- but every surface scene we see in the trilogy is as dark as a cloudy night; light coming only from the machines.  That wouldn't begin to support photosynthesis, which means all animal life would starve in, at most, a matter of a few years. How to finish current activity in Android Can't preview smoke simulation on Blender 2.82 Whenever I add a Quick Smoke effect, I can't preview it on the viewport unless I bake it.   And Yes, I added the Smoke Domain on the object.   I've seen tutorials like this  and this  that clearly shows the possibility of previewing the smoke before baking it.   I'm using Blender on a Mac with AMD GPU. I would guess that this is something related to OpenGL, as thoroughly explained here -  but that's for Windows.   Does anyone have any idea how to fix this? In contemporary American English usage, I come across sentences like:   I ain’t got no money.  Ain’t no man like him.    Saying ain’t no sounds incorrect to me because it is a double negative.  What is the origin of this particular construction? I wasn’t able to find anything online related to the origins of this incorrect usage. Explanation of finite population correction factor? I understand that when sampling from a finite population and our sample size is more than 5% of the population, we need to make a correction on the sample's mean and standard error using this formula:  $\hspace{10mm} FPC=\sqrt{\frac{N-n}{N-1}}$  Where $N$ is the population size and $n$ is the sample size.  I have 3 questions about this formula:   Why is the threshold set at 5%? How was the formula derived? Are there other online resources that comprehensively explain this formula besides  paper? Kaplansky's theorem of infinitely many right inverses in monoids? There's a theorem of Kaplansky that states that if an element $u$ of a ring has more than one right inverse, then it in fact has infinitely many. I could prove this by assuming $v$ is a right inverse, and then showing that the elements $v+(1-vu)u^n$ are right inverses for all $n$ and distinct.  To see they're distinct, I suppose $v+(1-vu)u^n=v+(1-vu)u^m$ for distinct $n$ and $m$. I suppose $n&gt;m$. Since $u$ is cancellable on the right, this implies $(1-vu)u^{n-m}=1-vu$. Then $(1-vu)u^{n-m-1}u+vu=((1-vu)u^{n-m-1}+v)u=1$, so $u$ has a left inverse, but then $u$ would be a unit, and hence have only one right inverse.  Does the same theorem hold in monoids, or is there some counterexample? Space war, humans recreated, grown from plants with an alien female supervisor This was a small hardback book I read in the UK in the late eighties. No memory of the cover.  Some kind of space war and a long lived alien female escapes the conflict in her starship, her side had lost.  An evil enemy that she needs to trap so she gets (I think) human  DNA from space dust and sets up on a habitable planet. The story then gets like episodic and looks at the world every couple of generations while she hides out there.  Plants bud on the world and people pop out from big seedpods, the population listens out for babies crying and rescue them.  There are legends of the 'Earth mother’ who secretly guides them to become high tech.   In reality she is creating attractive food bait for the bad aliens.  I think an early human and her are attracted to each other and she (maybe) gives him long life or (maybe) uses reincarnation to keep seeing him.  Edit:- the name 'Rimstalker' has just now popped up in my brain! Downgrade nVidia Shield to Android 4.4 KitKat How did this character come to be in the Nethersphere? How is it that       The Master   Came to be in the Nethersphere?  The last time we saw him      He died and refused to regenerate.  The Doctor even cremated his body.   So, given what previously happened to him, how was he now able to      Regenerate as Missy and take over the Nethersphere?