$\lim_{|P|\to0} \sum_{i=1}^n f(\xi_i)g(\eta_i)(t_{i}-t_{i-1})=\int_a^b f(x)g(x)dx$
Riemann and Darboux Integral of a product of two functions
Non-https links to other StackExchange sites in User profile when using https Showing diffeomorphism between $S^1 \subset \mathbb{R}^2$ and $\mathbb{RP}^1$ I am trying to construct a diffeomorphism between $S^1 = \{x^2 + y^2 = 1; x,y \in \mathbb{R}\}$ with subspace topology and $\mathbb{R P}^1 = \{[x,y]: x,y \in \mathbb{R}; x \vee y \not = 0  \}$ with quotient topology and I am a little stuck.  I have shown that both are smooth manifolds, and I used stereographic projection for $S^1$, but now I am runing into trouble when I give the homeomorphism between $S^1$ and $\mathbb{RP}^1$ as the map that takes a line in $\mathbb{RP}^1$ to the point in $S^1$ that you get when letting the parallel line go through the respective pole used in the stereographic projection.   If I use the south and north poles I get a potential homeomorphism, but I cannot capture the horizontal line in my image, but when I pick say north and east then my map is not well defined as I get different lines for the same point in $S^1$. Can somebody give me a hint how to make this construction work, or is it better to move to a different representation of $S^1$ ? How do I assign a value to a whole array of integers? General way to determine $\mathbb{Q}(\gamma) = \mathbb{Q}(\alpha,\beta)$ given $\alpha$ and $\beta$ Imputation before or after splitting into train and test? Preservation of ideals in localized rings Refer to localization of rings in Lang's Algebra p. 108. In particular, let $A$ be a commutative ring, $S$ a subset of $A$ that is a submonoid of the multiplicative monoid structure of $A$, containing the identity. Then  $S^{-1}A$ is the set of elements $\frac{a}{s}$ with $a \in A$ and $s \in S$.  We have $\frac{a_1}{s_1}=\frac{a_2}{s_2}$ if there exists $\bar{s} \in S$ such that $\bar{s}(s_2a_1 - s_1 a_2)=0$. Now consider the injection of $A$ into $S^{-1}A$ given by $x \mapsto \frac{x}{1}$. This is a ring homomorphism. Let $J$ be an ideal of $A$. Ring homomorphisms preserve ideals and so the injection of $J$ into $S^{-1}A$, call it $J^{*}$, will be an ideal. My problem is that i don't see why for any $\frac{a}{s} \in S^{-1}A$ and any $\frac{j}{1} \in J^{*}$ we will have that $\frac{aj}{s}$ will be in $J^{*}$, i.e. why there must exist an element $\kappa \in J$ such that $\frac{aj}{s}=\frac{\kappa}{1}$. Counting the number of surjections. How many functions from set $\{1,2,3,\ldots,n\}$ to $\{A,B,C\}$ are surjections? $n \geq 3$  Attempt  I was hoping to count the number of surjections by treating $A,B,C$ like bins, and counting the number of ways to fill them with $1,2,3,\ldots,n$ such that no bins are empty. This is equivalent to putting two "separator" bars in between the $n$ numbers or $n-1 \choose 2$.  However, I think I am missing a step here: prior to putting in the separator bars, I should permutate the $n$ elements. So, multiply by $n!$.  The order of elements in the same bins do not matter. So I should be divided by the number of permutations of the elements in each bins. This is where I am stuck.  Additional Info  I made the question up myself. Feel free to modify the question into a more solvable form if it is too difficult/complicated. Is it possible to build a .dmg file (for distributing apps) from a non-Mac platform? And if yes, how? How to change default user in WSL Ubuntu bash on Windows 10 I installed the WSL Bash on Windows 10 and set my user to e.g. 'abc' but now I want to change it to something else.  How can I change the user name of the user I'm logged in as?  When I open bash it automatically logs me in with my current user. Selecting the Real Analysis Textbooks I am a sophomore in US with double majors in mathematics and microbiology. I am interested in self-studying the real analysis starting now since it will help me on my current research on computational microbiology, prepare for upcoming math research (starting on this Fall) on the analytic number theory, and prepare for the real analysis course I will take on Fall and Putnam competition. I just finished "Calculus with Analytic Geometry" by G. Simmons, "How to Prove It" by Daniel Velleman, and "How to Solve It" by G. Polya. I also read some portions of Apostol's Calculu Vol.I to get a deeper view on the calculus theories. I was originally planned to read Apostol's Calculus Vol.I and Spivak's Calculus first, but I think it would be a better idea to start with real analysis since it covers all ideas in those "advanced calculus" textbooks and much more. My current plan is to start with one "dumbed-down" real analysis textbook and one "comprehensive, detailed, and intermediate" textbook, and advance into Rudin's PMA (required textbook for my real analysis course) starting on Summer and use it in accordance with other real analysis textbooks. Could you help me on selecting one book from each category?   Elementary Real Analysis textbooks:  **Elementary Analysis: The Theory of Calculus (Kenneth Ross)  **Understanding Analysis (Steven Abbott)  **The Way of Analysis (Robert Strichartz)  **Real Mathematical Analysis (Charles Pugh)   Intermediate, detailed Real Analysis textbooks:  **Mathematical Analysis (Tom Apostol)  **Undergraduate Analysis (Serge Lang)  **Introduction to Real Analysis (Bartle, Sherbert) **Elements of Real Analysis (Bartle, Sherbert)  **Mathematical Analysis I (Vladimir Zorich)   Thank you very much for your time, and I look forward to your advice!   Sincerely, PK I'm still new to ZFS. I've been using Nexenta but I'm thinking of switching to OpenIndiana or Solaris 11 Express. Right now, I'm at a point of considering virtualizing the ZFS server as a guest within either ESXi, Hyper-V or XenServer (I haven't decided which one yet - I'm leaning towards ESXi for VMDirectPath and FreeBSD support).   The primary reason being that it seems like I have enough resources to go around that I could easily have 1-3 other VMs running concurrently. Mostly Windows Server. Maybe a Linux/BSD VM as well. I'd like the virtualized ZFS server to host all the data for the other VMs so their data could be kept on a physically separate disks from the ZFS disks (mount as iscsi or nfs).  The server currently has an AMD Phenom II with 6 total cores (2 unlocked), 16GB RAM (maxed out) and an LSI SAS 1068E HBA with (7) 1TB SATA II disks attached (planning on RAIDZ2 with hot spare). I also have (4) 32GB SATA II SSDs attached to the motherboard. I'm hoping to mirror two of the SSDs to a boot mirror (for the virtual host), and leave the other two SSDs for ZIL and L2ARC (for the ZFS VM guest). I'm willing to add two more disks to store the VM guests and allocate all seven of the current disks as ZFS storage. Note: The motherboard does not have IOMMU support as the 880G doesn't support it, but I do have an 890FX board which does have IOMMU if it makes a huge difference.  My questions are:  1) Is it wise to do this? I don't see any obviously downside (which makes me wonder why no one else has mentioned it). I feel like I could be making a huge oversight and I'd hate to commit to this, move over all my data only to go fubar from some minute detail I missed.  2) ZFS virtual guest performance? I'm willing to take a small performance hit but I'd think if the VM guest has full disk access to the disks that at the very least, disk I/O performance will be negligible (in comparison to running ZFS non-virtualized). Can anyone speak to this from experience hosting a ZFS server as a VM guest? Motivation: I once had one surplus card on my hand, and knew the robbers would come soon. So I gave it to someone for free, to avoid losing more cards when they hit. I was surprised by how badly this was received. Two of the three other players (the hosts of the game) acted upset, like I had tried to cheat. I had to take the card back, get robbed the next turn and 'giving cards away for free' was banned immediately. (Oh, how I hate mid-game additions to the rules. But that's beside the point.)  So now the question: Is this really illegal in Settlers of Catan? Is it generally considered underhanded in any way? If so, why?   My reasoning was like this:        Consider players A and B.   A wants to give B a sheep.   A trades a sheep and one rock to B, in exchange for wheat.   A trades the received wheat for the (formerly his) rock.   A and B now both have the same resources they had in the beginning, only the sheep has moved from A to B. How to implement a follow author (or user) function? Is it possible to implement a Follow button for authors of posts in Drupal 7, so that users can:   Follow any one they wish, and Be notified about the new posts of those authors.   This sounds like 'Mashable Follow button' for authors.  Do you have any suggestions about how to implement this? Is Thomae's function Riemann integrable? Let $\displaystyle f: [0,1] \rightarrow \mathbb{R}$ given by   $$f(x) = \begin{cases} 0 &amp; x \notin \mathbb{Q} \\ \\ 0 &amp; x = 0 \\ \\ \frac{1}{q_x} &amp; x = \frac{p_x}{q_x} \in \mathbb{Q} \backslash \{0\}, \ p_x \in \mathbb{Z}, \  q_x \in \mathbb{N}, \ \text{gcd}(|p_x|, q_x) = 1 \end{cases}$$  Is $\displaystyle f$ Riemann integrable?  I am trying to use the equivalent statements $\displaystyle g:[a,b] \rightarrow \mathbb{R}$ is Riemann integrable and $\displaystyle \forall \epsilon &gt;0 \ \exists$ step functions $\displaystyle \rho, \psi$ with $\displaystyle \rho \leq g \leq \psi$ such that $\displaystyle \int_a^b (\psi - \rho) \leq \epsilon$.  I guess that means I would have to somehow show that given $\displaystyle \epsilon$, there exists only a finite amount of $\displaystyle x \in [0,1]$ with $\displaystyle f(x) \geq \epsilon$?   Is it recommended that I consider something else instead? If not, how should I do this? I'm trying to prove whether or not the Laurent polynomials $\mathbb{Z}[x, x^{-1}]$ with coefficients in $\mathbb{Z}$ form a principal ideal domain.    I know that $\mathbb{F}[x, x^{-1}]$ is a PID when $\mathbb{F}$ is a field, but clearly $\mathbb{Z}$ is not a field so I cannot appeal to this result. And my intuitions are not serving me very well at the moment. Can anyone provide a hint or direction to take? While doing some flagging (yes, I'm a little bit addicted) I stumbled upon the user . At first the case seemed clear, Spammer, but...and that's a big one, he seems to be a productive member of SU and SO (his account on SF isn't there anymore, but it's still in the Data-Dump of the explorer, not sure what happened).  He does never seem to link against StarWind-Software, except with his name and profile. He also  that he's a developer there...so...at what am I looking here? I can't really place him somewhere. Can't update drupal core to 7.27 How to know the model has started overfitting? Is there a known well ordering of the reals? So, from what I understand, the axiom of choice is equivalent to the claim that every set can be . A set is well ordered by a relation, $R$ , if every subset has a least element. My question is: Has anyone constructed a well ordering on the reals?  First, I was going to ask this question about the rationals, but then I realised that if you pick your favourite bijection between rationals and the integers, this determines a well ordering on the rationals through the natural well order on $\mathbb{Z}$ . So it's not the denseness of the reals that makes it hard to well order them. So is it just the size of $\mathbb{R}$ that makes it difficult to find a well order for it? Why should that be?  To reiterate:   Is there a known well order on the Reals? If there is, does a similar construction work for larger cardinalities? Is there a largest cardinality for which the construction works? Is there a specific term for the association of certain verb(s) with certain nouns (as direct objects)? Or vice versa. Or perhaps a name for the pair itself?  For example:   hold an opinion make a complaint offer an apology   Also, where might I find a list of such pairs?    EDIT: I chose my title wording randomly. I have no idea whether 'cognate verb' is the correct term for these things. However, I've just found out that 'cognate objects' are a defined class in linguistics, so perhaps my choice of words was inappropriate. I'm going to leave the title as it is, anyway.