If $M$ is the sigma algebra generated by $E$, then $M$ is the union of the sigma algebras generated by $F$ as $F$ ranges over all countable subsets of $E$.   (Hint: Show that the latter object is a sigma algebra)
Given a collection of sets $\mathcal{C}$ and $E$ an element in the $\sigma$-algebra generated by $\mathcal{C}$, how do I show that $\exists$ a countable subcollection $\mathcal{C_0} \subset \mathcal{C}$ such that $E$ is an element of the $\sigma$-algebra, $\mathcal{A}$ generated by $\mathcal{C_0}$?    The hint says to let $H$ be the union of all $\sigma$-algebras generated by countable subsets of $\mathcal{C}$....although I don't know why.
footnotes at the end of the book Is it possible to have all footnotes placed at one specific place in my document instead of bottom of each page? I have multiple footnotes scattered in the article and it would be clearer if all of them are at the very end of the article  Edit (moved from an ``answer'')  is it possible in endnotes to have links in each number where I placed footnote to the actual footnote at the end? Why do we focus so much in math on functions (as a subclass of relations)? Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and left-unique? Left- (or even right-) totality seem to be intuitive, since if an element doesn't appear in the domain, we might throw it out. But why left-uniqueness?  I'm looking for something like a "moral explanation" of why they would be the most useful subclass of relations.  My apologies if this is a previous question; I looked and didn't find much. Prove every odd integer is the difference of two squares I know that I should use the definition of an odd integer ($2k+1$), but that's about it.  Thanks in advance! Get relative path from comparing two absolute paths I want to leave my Mac awake overnight so it syncs 4GB of data from Dropbox. I need to prevent the machine from going to sleep because that pauses the Dropbox sync  I've been using  to keep the computer awake, but it also keeps the computer unlocked which is insecure.  I need a method to keep my Mac awake (so that Dropbox can sync overnight) but also locked (so no one can read my email). It'd be great if this method also turned off the screen (to preserve power), but honestly that's not even 100% necessary. I have a manuscript that is under review in a major journal which is a main result out of a few recent years of my research. When applying for faculty positions, I would like to send the draft as a part of my writing sample. But since the material is not yet published, I am not sure if it is a good idea.  What are the foreseeable disadvantages of sending an article in review to the faculty search committee?  My area is neuroscience / biomedical engineering.  Update: The paper got into a high-profile journal, and I got a job at a different place which did not require a writing sample. I am unable to figure out simple solution to add "Bookman Old Style"  as a main font in my latex document. I have used below line but does not give me required font.  \usepackage{bookman}   Could not find any Simple correct answer for this. Pl. suggest how to do it.  I am using windows, pdflatex.  thank you Holomorphic function has at most countably zeros Let $f$ be holomorphic in a domain $U$, then $f$ has at most countably zeros.  How can I prove this statement? I have seen it in Sheng Gong  explicitly, but without any indications. Creating label in QGIS? I am using QGIS 2.18.16 on Windows 10.    What I want is a label to appear within the boundaries of a polygon.    I select a layer, then select command Layer>Labelling, and I get:    From here, I can change "No Labels" to "Show labels for this layer".  But I can't actually create label text.  I find a clue in a prior question:    “Labelling polygon using QGIS?”    The answer was:  The layer you are trying to label must have at least one attribute field populated with data. You can then turn on labelling in Layer Properties and select which attribute field to use for your labels.  I can enter command Layer>Open Attribute Table, and what I get is:    If I need something in the attribute table in order to apply a label, how do I accomplish that? I did some calculation and found out, that my Living End deck (modern) might be better of with 66 cards maindeck. SB still contains of 15 cards. Depending on the match up, I'd like to cut my deck for game two down to the normal 60. The SB than would contain 15+6 = 21 cards. Is this legal? Does Thunderbolt support chaining multiple monitors to a MacBook Pro? This has always been a big want for me. I love using multi-monitor setups but only want a single laptop. Does the 2011 MBP Thunderbolt support chaining multiple monitors together? I know that you can add a monitor and say, a few hard drives, but can you connect it to 2 27" monitors and your flat screen TV (for example)? Thanks! My website title shows up slightly different from what's in my title tag, only in Google search The following question is from Artin's Algebra.      If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions.    I have no idea on how to tackle this question. I tried block multiplication, but it didn't appear to work. finite elements-exercice We consider in $\mathbb{R}^2$ the set of points $$\{M_1(-1,1),M_2(0,1), M_3(2,1),M_4(-1,0),M_5(1,0),M_6(2,0)\}$$  Let $\Omega$ a rectangular structure consisting of the heads $\{M_4(-1,0),M_6(2,0), M_3(2,1),M_1(-1,1)\}$  Let the two polygons $\sum_1 = \{M_4,M_5,M_2,M_1\}$ and $\sum_2 = \{M_5,M_6,M_3,M_2\}$   let the mesh $\{(e_1,\sum_1,Q_1),(e_2,\sum_2,Q_2)\}$ such $$Q_1=\{f : \mathbb{R}^2 \rightarrow \mathbb{R} , f(x,y) = a + bx + cy + d xy , a , b, c , d \in \mathbb{R}\}$$  1- Prove that $(e_1,\sum_1,Q_1)$ and $(e_2,\sum_2,Q_1)$ are two finite elements of Lagrange.  2- Let $V$ the finite-dimensional space that corresponds to the mesh. Prove that $V \nsubseteqq H^1(\Omega)$  My problem is to prove 2. US / Mexican border data I'm interested in locating free GIS shapefile data representing the US / Mexican border. I have found a point layer depicting the border crossing points but I'm interested in a polyline feature representing the actual border. Intermediate ring between a field and an algebraic extension. This is an exercise in some textbooks.     Let $E$ be an algebraic extension of $F$. Suppose $R$ is ring that contains $F$ and is contained in $E$. Prove that $R$ is a field.   The trouble is really with the inverse of $r$, where $r\in R$. How to prove that $r^{-1}\in R$, in apparent lack of a characterization of $R$.  It occurred to me to use the smallest field containing $R$ ($R$ is easily shown to be an integral domain), that's the field of quotients, and proving that it's $R$ itself, but I don't really know how to proceed.  A not-too-weak, not-too-strong hint will be much appreciated.   Beware $ $ Readers seeking only hints should  beware that there is now a complete answer. Suppose $G$ is a group and has exactly two nontrivial proper subgroups. Prove that $G$ is cyclic and $|G|=pq$ where p,q are distinct primes or $G$ is cyclic and $|G|=p^3$ where $p$ is a prime. Usually my tactic to show a group is cyclic is find an elements such that its order and order of the group are the same. But here I cannot do so because I have no informatio regarding order of elements. So I stuck. Can anyone help me? Probability of first actor winning a "first to roll seven with two dice" contest? Two players P and Q take turns, in which they each roll two fair and independent dice. P rolls the dice first.  The first player who gets a sum of seven wins the game. What is the probability that player P wins the game? I would like to know if it is possible to enable free shipping with Woo Commerce only if certain items (or items of certain categories / shipping classes) are included in the cart.  The Free Shipping option only activates on the basis of cart value or via a coupon code so I investigated Flat Rate shipping and discovered that it prioritises the most expensive shipping class over the cheaper ones. This means the free shipping class I created never overrides the default charge.  I understand the logic behind this but it's the opposite of what I need as the idea is to incentivise purchases of higher profit margin products by eliminating delivery charges when they're included in an order.  I don't know if what I am trying to achieve is simply not possible within Woo Commerce. A coupon that only applies to certain products / categories seems to be the closest I can get but I'd rather not force users to enter a code at checkout.  Does anybody know if there any simpler ways to give users who purchase certain products free shipping? Why won't my villagers breed I have created the following structure. The intention is to create fairly compact houses for villager breeding.   This is a section of the houses.   I have 2 of these structures, which would resolve to 28 doors and 4 other types of houses meaning I have 32 houses in total. healed healed 3 villager zombies and locked them inside a small house. One of them sometimes shows hearts, but they won't breed.   I am playing vanilla Minecraft 1.8. The doors are spruce.  What is wrong with these houses? Is there something wrong with spruce doors or half slab roofs?