comparison of mean a subpopulation with entire population
Compare means of group with a mean of a subgroup
Prove holomorphism of expression I have to prove that given $\gamma$ contour of $G \subset \mathbb{C}$  and $f$ continuous function on $G$. Then the function $g(z) = \int_{\gamma} \frac{f(w)}{w-z}dw$ is holomorphic on $\mathbb{C} - \bar{G}$ My attempt: I started by using that given any parametrization $\alpha :[a,b] \rightarrow \gamma$ we can rewrite $g(z)$ as: $g(z) = \int_a^b \frac{f(\alpha(t))}{\alpha(t)-z}\alpha'(t)dt$ Now using the fundamental theorem I derivate to see where $g'(z)$ is defined and I get: $g'(z) = \frac{f(\alpha(b))}{\alpha(b)-z}\alpha'(b) - \frac{f(\alpha(a))}{\alpha(a)-z}\alpha'(a) $ Here I get stuck, I fail so see how this links back to $g$ being holomorphic at $\mathbb{C} - \bar{G}$. Any hints on how to proceed? Thanks! How do I enter a strong (long) DKIM key into DNS? How to boot from non-Ubuntu live ISO images like Fedora or CentOS? How to share worlds between platforms with Minecraft Better Together I have read that with the new "Better Together Update" (BTU) it is now possible to .  However, I am having trouble finding details of how this is supposed to work to determine if it would do what I am hoping it would and therefore whether it is worth my time/money.  I have some saved worlds on Minecraft for XBox One, all done in creative mode.  I am able to import them into what is now just called "Minecraft" which as of the BTU is what is being used to mean the cross-platform version.   Can I now play this same world on my Windows 10 PC?   If I go to the Minecraft web site it offers to sell me the "Java Edition" with the addendum "includes edition for Windows 10".   But from what I have read, anything that has the word "edition" in it isn't the cross platform version.   It's very unclear to me what I would actually be buying here.  If it is possible to open the same world on different platforms, exactly how would I go about doing this?  On the XBox I have to sign in with an outlook.com e-mail address to access my worlds, or my kids' sign-ins to access their worlds, is it the same from the PC? And where would my world be saved?   I have read that XBox saves will be in the cloud, but PC saves will be local, so how can I get a cloud save onto my PC?  And then how can I get it back to the cloud when I am done so I can play it on the console again?  Will there be two copies floating around, making it possible for me to make a backup on my PC?  How will this interact with world editors?  At some point will I be able to start a game on XBox, transfer it to the PC, open it in a world editor, make some changes, and then transfer it back to the XBox? I've recently started maintaining someone else's JavaScript code. I'm fixing bugs, adding features and also trying to tidy up the code and make it more consistent.  The previous developer used two ways of declaring functions and I can't work out if there is a reason behind it or not.  The two ways are:  var functionOne = function() {     // Some code };     function functionTwo() {     // Some code }   What are the reasons for using these two different methods and what are the pros and cons of each? Is there anything that can be done with one method that can't be done with the other? Cannot find Underground Jungle Temple I know classical physics, quantum mechanics, special relativity, and basic nuclear physics. I would like to get into some particle physics.   Where can I find a good introduction? It might be useful to segment books by whether they require quantum field theory or not. Say, for example, you and a group of people were all sailing out in the ocean and something happens...then you say,  "I guess we are all in the same boat"  You are literally in the same boat with everyone else, and your current situation is the same as everyone else.  Now such a phrase can be applied literally and figuratively. I was wondering if there was a word that would mean both literally and figuratively. I know this is a tad bit of an oxymoron, but I was curious to see if there was such an "all encompassing" word for this. Word for person who looks for flattery What word could be used to describe a person who seeks and seems to depend on constant positive feedback from the people around him. The problem:  Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t) - f(x)| \leq |t - x|^2$ for all $t, x$. Prove $f$ is constant.  I believe I have some intuition about why this is the case; i.e. if $t$ and $x$ are very close, ($|t - x| = \epsilon$), then $\sqrt{\epsilon}$ will converge to 0, so $|f(t) - f(x)|$ will also converge to zero if you keep taking $t$ and $x$ closer and closer to each other. However, how do I formalize this argument? Thanks. Beamer: Vertical alignment of multi-column ToC For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives. Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset $K\subseteq \Omega$. Then by Cauchy Theorem and Morera Theorem, $f$ is holomorphic. Let $f_n$ and $f'$ be the derivatives of $f_n$ and $f$ respectively. Prove that $f_n'\to f'$ uniformly on any compact subset $K\subseteq \Omega$.   How to prove? Beamer keeping line position constant on itemize I want to have a slide in beamer with itemize exposing new line on each click while at the bottom part of the slide there are some equations. This is my attempt  \begin{frame}   \begin{overprint}{\textwidth}     \leavevmode     \begin{itemize}[&lt;+-&gt;]     \item &lt;1-&gt; Poisson Boltzman equation (PB)     \item &lt;2-&gt; Debye-H\"uckle equation  (DH)     \item &lt;3-&gt; The naive approach - applying (DH) equation to PE     \end{itemize}   \end{overprint}   \begin{tikzpicture}     \node at (0,0) {};         \node&lt;1,2&gt; [text width = 20cm]at (10.5,-1) {$ \nabla^2 \phi =\sum_i \frac{n_i q_i\text{e}^{-q_i \phi/kT}}{4 \pi \epsilon}$};         \node &lt;2&gt; [text width = 20cm]at (10.5,-2) {$q_i \phi/kT \ll 1 \Rightarrow \nabla^2 \phi \simeq \kappa^2 \phi$};         \node &lt;2&gt; [text width = 20cm, font=\footnotesize]at (10.5,-2.5) {$\kappa^2 = \lambda^{-1/2} =  \sum_i \frac{n_i q_i}{4 \pi \epsilon}$};         \node &lt;3&gt; [text width = 20cm] at(10.5,-1) {$\frac{F_{el}}{VkT} = - \left ( \frac{ 4 \pi e^2}{\epsilon kT}     \sum_{i=1} n_i z_i^2 \right )^{3/2} \left ( 12 \pi \right )^{-1}$};   \end{tikzpicture} \end{frame}   However, the itemized line jump up and down between slides. I've tried to use itemize without any environment surrounding it. I've also tried to use overlayarea instead of overprint. How can I keep the line position constant?  I've uploaded a gif animation to display this. On my machine it run really fast and I can't fix this. Hope the problem is understood.   When I run this, I get Suppose that $f:M \to N$ is a smooth map of two manifolds and $E$ is a vector bundle over $N$. Given a connecton $\nabla$ on $E$ one can pull it back to $f^*N$ (which is a bundle over $M$) and obtain a connection $f^* \nabla$.   If $E=TN$ is a tangent bundle one can consider the Levi-Civita connection $\nabla$, with respect to the chosen metric $g$ on $N$. It is characterised uniquely as torsion free metric connection. Given a map $f:M \to N$ one can pullback the metric $g$ to $f^*g$. If it happens that $f^*g$ is again a metric (which is the case for example for immersions) one can consider the Levi Civita connection on $M$ with respect to $f^*g$. I wonder how these two constructions are related, in particular      Is there a way to pullback affine connections to get again affine connection (i.e. connections on tangent bundle)?     Additionally I would like to know:     What is the relation between Christoffels symbols for the connection and for its pullback? Convergence of the series $\sum_n \frac{n^{\log(n)}}{\log(n)^n}$ I have a rough outline of a proof. Is this correct?  $$\sum_{n \geq 2} \frac{n^{\log(n)}}{\log(n)^n} \to a \in \mathbb{R} \iff  \sum_{n \geq 2} 2^n \frac{(2^n)^{\log(2^n)}}{\log(2^n)^{2^n}} \to b \in \mathbb{R}$$ via dyadic criterion (or Cauchy condensation test).  Analyzing the right-hand side:  $$\sum_{n \geq 2} 2^n \frac{(2^n)^{\log(2^n)}}{\log(2^n)^{2^n}} = \sum_{n \geq 2} 2^n \frac{(2^n)^{n\log2}}{(n\log2)^{2^n}} = \sum_{n \geq 2} \frac{(2^n)^{n\log2 + 1}}{(n\log2)^{2^n}} = \sum_{n \geq 2} \frac{2^{n^2\log2 + n}}{(n\log2)^{2^n}} = (\ast). $$  We know for some sufficiently large $N$, $n \log 2 \ge 2$ for all $n\ge N$, so we can bound the series above by:  $$(\ast) \leq C_N + \sum_{n \geq N} \frac{2^{n^2\log2 + n}}{2^{2^n}} = \sum_{n \geq N} 2^{n^2\log2 + n - 2^n} = (\ast\ast),$$  where $C_N$ is the finite sum of the original series until $N$. And once again, for sufficiently large $M$, $m^2\log2+m - 2^m \leq -m^2$ for all $m \ge M$. Hence  $$ (\ast\ast) \leq C_{\max(N,M)} + \sum_{n \geq \max(N,M)} 2^{-n^2} &lt; \infty.$$  We then use the $p$-test to conclude that the sequence does converge. How are NFL matchups determined for each season? During the 2012 NFL Season, the Pittsburgh Steelers and Baltimore Ravens play each other in weeks 11 and 13, two weeks (in both NFL schedule and time) apart. To me, that seems too close for comfort.   I do know that teams play other teams in their division twice. How does the NFL schedule procedure work? When in Journey Builder what is the difference between journey data and contact data? I am creating a JB with an engagement split. What is the difference between Contact Data and Journey Data. ie what will I see listed in the two groupings. If I choose the journey data will this only be as up to date as the time that the data entered the Journey? Book: Handful of children kept in a school as political mythological prisoners Unity remove logo in splash screen Notation: What does the tilde below of the expectation mean?