Can't copy a DataTable
What is a NullPointerException, and how do I fix it?
Sum of two squares modulo p I have heard somewhere that for all primes $p$, for all $k$, there exist $x, y$ s.t. $x^2 + y^2\equiv k \pmod{p}$? I recall that the proof is very elementary, but I can't remember such a proof. How would one prove this? One way is to use Cauchy-Davenport, but I don't think that this is the simplest solution. Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$ Recently, I ran across a product that seems interesting.  Does anyone know how to get to the closed form:  $$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$  I tried using the identity $\cos(x)=\frac{\sin(2x)}{2\sin(x)}$ in order to make it "telescope" in some fashion, but to no avail. But, then again, I may very well have overlooked something.  This gives the correct solution if $n$ is odd, but of course evaluates to $0$ if $n$ is even.   So, I tried taking that into account, but must have approached it wrong.   How can this be shown? Thanks everyone. Finitely generated group which is not finitely presented Is there any easy group theoretical way of showing that the wreath product $G$ of two infinite cyclic groups is not finitely presented?  I was looking for a finitely presented group with a central subgroup isomorphic to the free group of countable rank and whose factor group is isomorphic to $G$; in this way assuming that $G$ is finitely presented we get a contradiction.   Any ideas? Best way to compare 2 XML documents in Java I'm trying to write an automated test of an application that basically translates a custom message format into an XML message and sends it out the other end.  I've got a good set of input/output message pairs so all I need to do is send the input messages in and listen for the XML message to come out the other end.  When it comes time to compare the actual output to the expected output I'm running into some problems.  My first thought was just to do string comparisons on the expected and actual messages.  This doens't work very well because the example data we have isn't always formatted consistently and there are often times different aliases used for the XML namespace (and sometimes namespaces aren't used at all.)  I know I can parse both strings and then walk through each element and compare them myself and this wouldn't be too difficult to do, but I get the feeling there's a better way or a library I could leverage.   So, boiled down, the question is:  Given two Java Strings which both contain valid XML how would you go about determining if they are semantically equivalent?  Bonus points if you have a way to determine what the differences are. Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal, but that the set of finite sums of products of elements of I and J is an ideal. This ideal is called the product ideal, and is denoted by IJ.   I want to check if my work on the first part makes any sense. I am not sure if I am picking the right example.  If I let R = Z and I = (2) and J = (3) then I has multiples of 2, and J has multiples of 3  6 will divide the product IJ if we add 2 and 3 though we get 5 in IJ, but I am not sure if 5 Divides IJ. I'm a bit overwhelmed by the available options in regards to the three major photo editing apps from Adobe.  Does anyone know of an article or site that lists the features of Lightroom, PS Elements, and PS, side by side?  I know that Lightroom revolves around edits and workflow, and that PS is the big kahuna of image editing.  Where does Elements fit in to this understanding? Intuition for Smooth Manifolds Consider the graphs of the functions $f_1(x) = |x|$, and $f_2(x) = x$ under the subspace topology of $\mathbb{R}^2$.   Both of these graphs are smooth manifolds, just pick coordinate charts to be $(x, f_i(x)) \leftrightarrow x$.   Moreover, they are diffeomorphic via the map $(x, f_1(x)) \rightarrow (x, f_2(x))$.   This seems to clash with my intuition. For example, the graph of $f_1$ has a corner, so it "shouldn't" be smooth, much less diffeomorphic to $f_2$, which is just a straight line.   Can someone explain what's going on here?   In light of these examples, how should I visualize smooth manifolds and diffeomorphisms? Network button missing gone from Nautilus? I'm using Ubuntu 16.04.3 LTS.  I remember having a link in Nautilus that I used to browse network shares with in the past. I don't remember exactly but I think it was "Windows network" or "Browse network".  Now I can't find this on my current installation, there's only "Connect to server", but this is not what I want. Is there a way to put that link there? How to redirect output of a program to a ZIP file? Seeing through objects in Game engine Every real number in $[0,1]$ has a decimal expansion $0.d_1d_2d_3...$, so construct an infinite tree rooted at 0 where each node has branches leading to $\{0,1,2,3,4,5,6,7,8,9\} $, and let each path through the tree represent the successive decimal digits of a real number. To enumerate them, go through the tree breadth-first i.e. layer-by-layer. Prove that if $P$ is a $p$-Sylow subgroup of $G$ and $N \triangleleft\&gt; G$ then:    $ PN/N $ is a $p$-Sylow subgroup of $G/N$ $P \cap N $ is a $p$-Sylow subgroup of $N$ Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$. Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G   has subgroup of order $\operatorname{lcm}(n,m)$.   This is a statement that my lecturer mentioned in my (beginners') Abstract Algebra class. I'm not sure I understand why it's true.  What I have so far: Use the abelian group structure theorem on $\langle H, K\rangle$ (finite group generated by $H$ and $K$). Then $\langle H,K\rangle=C_{a_1}C_{a_2}\dotsm$ and $n, m |\langle H,K\rangle$. Which means it's also true that $\operatorname{lcm}(n,m)|\langle H,K\rangle$. Can I leverage this to say there's a subgroup of order $\operatorname{lcm}(n,m)$? Why is the gate charge curve (Miller plateau) of MOSFETs dependent on Vds? For fixed $z_i$s inside the unit disc, can we always choose $a_i$s such that $\left|\sum_{i=1}^n a_iz_i\right|<\sqrt3$? Let $z_1,z_2,\ldots,z_n$ be complex number such that $|z_i|&lt;1$ for all $i=1,2,\ldots,n$. Show that we can choose $a_i \in\{-1,1\}$, $i=1,2,\ldots,n$ such that $$\left|\sum_{i=1}^n a_iz_i\right|&lt;\sqrt3.$$ Destroying a Planewalker before its ability is used Say my opponent plays a planeswalker during their first main phase.  I have a  or some other effect that will kill it after it is already in play, but before they use any of its abilities.  I could respond to them using a +1 ability, but I want to kill it before that effect goes on the stack.  In most casual situations, it would basically come down to whether I say "Hero's Downfall" first, or they say "+1" first.  But I can't imagine that's how it's supposed to work.  Is there a time between when the planeswalker hits the battlefield and when they activate its abilities that they must let me respond? It seems to me that situations like this arise frequently, and they are resolved simply by who speaks up first, which is horrible for such a rules-oriented game. Let $P$ denote the quotient space obtained by the action of $\mathbb{Z}\backslash2\mathbb{Z}$ by the antipodal map $z\mapsto\frac{1}{z}$ on the riemann sphere $\hat{\mathbb{C}}$ (identified here with $\mathbb{C}\cup\left\{\infty\right\}$). I identify $P$ with the set:  $\left\{ z\in\mathbb{C}:\left|z\right|&lt;1\right\} \cup\left\{ e^{it}:0\leq t\leq\pi\right\} $   that is to say, I use elements of the above set as the representatives for the equivalence classes in $P$. I am looking for formula for a function $f:P\times P\rightarrow[0,\infty)$ such that $f$ is a metric on $P$. Specifically, I would like a formula for $f$ that I can evaluate by plugging in representative elements in the above set (or something like that, more or less).  Thanks in advance.  Edit: forgive me for sounding desperate, but I cannot make due with an explanation of how to obtain such a formula. I want the formula.   An analogy for you if I have yet to make myself clear: suppose I was asking for the area of a square with side length $s$. The answers I have received so far for my question are akin to saying "multiply $s$ by itself" or "use the area formula for a square". The answer I am looking for is akin to saying "$s^{2}$". I need the formula. And please, no expressions with differentials, nor matrices, or any of that. I need to know how to compute the metric by using the complex numbers in the indicated set that I have identified with $P$. The function $f(z)=z+\frac{1}{z}$ maps $\Omega$ biholomorphically on the upper plane The question states:     Let $\Omega$ be the upper half plane minus the unit disk. Show that the map $f: z \mapsto z+1/z$ maps $\Omega$ biholomorphically on the upper plane. Describe the preimages of the lines $\mathrm{Im}(z)=\text{a constant}&gt;0$.   So how exactly does one show the mapping is biholomorphic. I'm not sure where I should start, so unfortunately I haven't made any progress to Global Variables inside functions I think that all the zeros of the Riemann-Zeta function ${\zeta}( z ) = \frac{1}{1-2^{1-z}} \sum_{n = 0}^{\infty} \frac{1}{2^{n+1}} \sum_{k = 0}^{n} (-1)^k \binom{n}{k} (k+1)^{-z}$  have real part smaller than 1, but I don't know how to prove it. If Re(z) > 1 than  ${\zeta}( z ) = \sum_{n = 1}^{\infty} \left( \frac{1}{n} \right)^z$