How to prove that $1^r + 2^r +... + n^r = a_1 n^1 + a_2 n^2 +... + a_{r+1} n^{r+1}$?
Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula
Iterating each character in a string using Python In C++, I can iterate over an std::string like this:  std::string str = "Hello World!";  for (int i = 0; i &lt; str.length(); ++i) {     std::cout &lt;&lt; str[i] &lt;&lt; std::endl; }   How do I iterate over a string in Python? How do you make an image emission without washing out the image? I am trying to make a computer monitor. This means I have to have an image texture for the screen. My image texture is currently connected to an Emission BSDF shader. Unfortunately, when the emission is bright enough to emit light, the image is white-washed. I tried adding a Brightness/contrast node to the image before pushing it into the Emission BSDF, but it seems to directly counter the Emission's strength.  The desired light level   The desired image brightness Tracking disk I/O by application in Windows? Is there a utility for Windows that operates similarly to iotop for Linux?  I'm looking for simple tool that can tell me what applications are writing to disk and how much they're actually writing. How do I determine my wheel size Would someone explain wheel sizes.  If I am not mistaken, the wheels on my Trek 520 touring bike are 700C. There are no markings, and of course, no documentation of any kind. I assume 700 means 700mm, so 700 / 25.4 = 27.56 inches.  If I measure the rim however, its diameter is approximately 25.1 inches.  If I buy Continental tires, the size is 28 inch, but I have to buy 27 inch Bontrager tubes.  To make matters worse, per this link, tires come in sizes of 26, 27, and 28 inch, and size 700.     How can I determine my wheel size, and are these various sizes interchangable. Also, is the 'C' suffix important, and what does it mean?  Of course the tires are marked, but I am under the impression (per the dealers comments) that Continental marks their tires differently than everyone else. Also, the tire markings don't help for ordering a new wheel.   With so many tire sizes available, how can I know if I am using the correct size.  After reading the various answers here, I should explain that my confusion on this matter stemmed from the fact that I used to work as an auto mechanic. In the auto industry, car tires are based on wheel size, 13", 14", 15" etc, and there is no possible way to mount a 15" tire on a 13" rim for example.  So when I saw all these different tire sizes, I assumed they all required a different wheel size, but in fact, there is essentially just one wheel diameter for my type of bike, which they refer to as 700C, which is actually 622mm.   I also initially assumed that the C referred to the wheel's width, but that isn't true either. The C is apparently just antiquated terminology.  So at least in theory, I can mount a 26", 27" 28" or 29" tire on my 700C (622mm) rim. I just need to verify the width of the tire is suitable my wheel width, which is what Sheldon Brown's article explains.  Also, I measured my Continental tires, which are marked as 28 inch. They actually measure 27 inches (fully inflated), which must be why my dealer calls their sizing non standard. Complex numbers equation: $z^4 = -16$ Latest NVIDIA driver on Ubuntu 16.04 Using a Single system() Call to Execute Multiple Commands in C In an information security lab I'm working on, I've been tasked with executing multiple commands with a single call to "system()" (written in C, running on Fedora).  What is the syntax that will allow me to execute more than command through system()?  (The idea being you could execute arbitrary commands through a program running on a remote computer, if the program interacts with the OS through the system() call.)  I.e.:  char command[] = "????? \r\n";  system(command); How to print colored text to the terminal? How can I output colored text to the terminal in Python? Checking the way to stop duplicate names - validation rules or triggers Show that for $\eta > 0$, $\alpha_\eta=-\infty$ and for the Cauchy problem $z'=x-z^2,z(0)=\eta$ the saturated solution is defined up to $+\infty$. We consider the differential equation $y'=x+y^2$ and for any $\eta \geq 0$ we denote by $\phi_\eta:(\alpha_\eta,\beta_\eta)\rightarrow\mathbb{R}$ the unique saturated solution of the Cauchy Problem $y'=x+y^2$, $y(0)=\eta$. I am required to do the change of variable $z(x)=y(-x)$ and I succeded here, obtaining $y'(-x)=y^2(-x)-x$ and so $z'(x)=x-z^2(x)$, then I was required to represent the $0-isocline$ and it was also simple as I obtained a parabolla and also I was required to write the equation of the tangent in the point $(0,\eta)$ which was also pretty simple as I obtained $\phi'_\eta(0)=\eta^2$ and the equation of the tangent is $y=\eta+\eta^{2}x$. Now, I am required to find $\phi^{(5)}_0(0)$( that $5$ indicates the number of differentiations), which puzzles me as I don't know the form of the solution and if it can be found out and also I am told to show that $\frac{1}{\eta}\geq\beta_\eta$ What I did not prove after the first edit : and that for $\eta &gt; 0$, $\alpha_\eta=-\infty$ and for the Cauchy problem $z'=x-z^2,z(0)=\eta$ the saturated solution is defined up to $+\infty$. At this last ones I am lost. Any help, please? Edit: I have passed beyond finding $\phi^{(5)}_0(0)$ and I think I did show that $\frac{1}{\eta}\geq\beta_\eta$, now it remains to prove the last questions. R version 3.1.1 (2014-07-10) -- "Sock it to Me" &gt; bl &lt;- c(140, 138, 150, 148, 135) &gt; fu &lt;- c(138, 136, 148, 146, 133) &gt; t.test(fu, bl, alternative = "two.sided", paired = TRUE) Error in t.test.default(fu, bl, alternative = "two.sided", paired = TRUE) :  data are essentially constant   Then I change just a single character in my fu dataset:  &gt; fu &lt;- c(138, 136, 148, 146, 132)   and it runs...  &gt; t.test(fu, bl, alternative = "two.sided", paired = TRUE)      Paired t-test   What am I missing here? Translation of these Cirth runes What is the translation of what's written below?  I think it may be the poem of the One Ring, but I'm not sure. Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as the dual space of $E$.($\dim E=n$)   $\Omega^P(E):=\{ \alpha\colon \overbrace{E\times\cdots\times E}^{p- times}\rightarrow \mathbb R\  \ , \alpha \text{ is alternating multilinear map}\}$ $\Omega^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\  \ , u \text{ is alternating multilinear map}\}$  I wish to prove that there exist an $\textbf{unique}$ bilinear form $B\colon \Omega^p(E^*)\times\Omega^p(E)\rightarrow\mathbb R$   such that for all $ f_i\in E^* \ , u_j\in E^{**}\simeq E \qquad  i,j=1,\cdots,p$, it has the following rule:  $B(u_1\wedge\cdots\wedge u_p,f_1\wedge\cdots\wedge f_p)=\det[f_i(u_j)]$.  $\wedge$ is wedge product between alternating maps i.e. $\alpha\wedge\beta(v_1,\cdots,v_{p+p})=\dfrac{1}{p! p!}\displaystyle\sum_{\sigma\in S_{p+p}}sign(\sigma)\alpha(v_{\sigma(1)},\cdots,v_{\sigma(p)})\beta(v_{\sigma(p+1)},\cdots,v_{\sigma(p+p)})$  How can I do this? MLE for Uniform $(0,\theta)$ I am a bit confused about the derivation of MLE of Uniform$(0,\theta)$.  I understand that $L(\theta)={\theta}^{-n}$ is a decreasing function and to find the MLE we want to maximize the likelihood function.  What is confusing me is that if a function is decreasing, then wouldn't the function be maximized at the smallest input rather than the largest?  Thank you in advance for your help. Make bash exit != 0 when called by AWK and interrupted with ^C I'm having an issue with the following piece of (G)AWK script:  do {     ... } while (system("sleep 10"))   My intention is to break the loop when the user presses ^C during the sleep, but it's not working.  I believe the problem is that Bash exits with 0 when interrupted with ^C, at least when it's executed by AWK's system():  $ awk 'BEGIN { print "\n" system("sleep 2") }' (let the sleep complete) 0      $ awk 'BEGIN { print "\n" system("sleep 2") }' ^C 0   Why is it so?   Is this a bug in Bash or in (G)AWK?  Is there a simple solution that doesn't involve complicated Bash-specific syntax like trap?  The best I could come up with is this:  do {     ... } while (42 == system("sleep 10 &amp;&amp; exit 42"))   which still looks like a kludge to me. Which is correct: "This is her" or "This is she"? Upon answering the telephone, the person calling asks if Joan is available. If Joan is the person who answered the phone, should she say "This is her" or "This is she"? If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$ an ideal, then $M$ has a structure of $R/I$-module Let $M$ be an $R$-module and let $I$ be an ideal of $R$ such that $I$ is a subset of $\mathrm{Ann}(M)$. Define a product of an element of $R/I$ by an element of $M$ as follows: $$(r+I)\cdot m=rm.$$ (a) Show that this product is well defined. (b) Show that this product, together with the sum of elements of $M$, turns $M$ into an $R/I$-module.  My answer for (a): Let $r+I,s+I \in R/I$ such that $r+I=s+I$. Then $(r+I)m=(s+I)m$, for all $m \in M$. Then $rm=sm \implies r=s$  (since the inverse of $m$ is in $M$), so the product is well defined.  Is my answer to (a) correct?  How can I use Ann(M) to answer (b)? How does the Eco-Cooler air conditioner really work? describes a device for developing countries which apparently cools the air by 5 °C without electricity. According to , it works by lowering the pressure to drag in the ambient air. Wouldn't the ambient air be at the same temperature?  Does the constriction compress the air and temporarily increase the pressure and so temperature, which then dissipates heat because it is hotter than surroundings; then after passing the constriction, the pressure returns to almost ambient but the temperature drops to below ambient?  How does the Eco-Cooler air conditioner actually work, if at all? If $f:U\to V$ is holomorphic and $f'(z)\neq 0$ for all $z\in U$, then$f$ is locally bijective. I am trying to solve the following problem:     Let $f:U\to V$ be a holomorphic function such that $f'(z)\neq 0$ for all $z\in U$. Show that for all $z_0\in U$, there exists a disc $D_\varepsilon(z_0)\subseteq U$ such that $f:D_\varepsilon(z_0)\to f(D_\varepsilon(z_0))$ is bijective.   Atempt:  I am trying to use Rouche's Theorem, but I am stuck on a particular step. Let $z_0\in U$. Since $f'(z_0)\neq 0$, there is a disc $D_r(z_0)\subseteq U$ such that $$f(z)=f(z_0)+(z-z_0)h(z),\quad\forall z\in D_r(z_0)$$ where $h$ is holomorphic on $D_r(z_0)$ and $h(z)\neq 0$ for all $z\in D_r(z_0)$.   Let $0&lt;\varepsilon&lt;r$ to be defined later and fix $w\in D_\varepsilon(z_0)$. To show that $f$ is injective in $D_\varepsilon(z_0)$ is to show that the function $f(z)-f(w)$ has exactly one zero in $D_\varepsilon(z_0)$. But $$F(z):=(z-z_0)h(z)$$ has exactly one zero in $D_\varepsilon(z_0)$, so we might want to apply Rouche's Theorem with $F$ and $$G(z):=f(z)-f(w)-(z-z_0)h(z)=f(z_0)-f(w)=-(w-z_0)h(w)$$ to conclude that $F$ and $F+G$ have the same number of zeros and hence $f$ is injective. But that means that we need to find $0&lt;\varepsilon&lt;r$ such that $|G(z)|&lt;|F(z)|$ on $\partial D_\varepsilon(z_0)$. That is, $$|w-z_0||h(w)|&lt;|z-z_0||h(z)|$$ for all $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$.  Can we find such $\varepsilon&gt;0$?  It seems intuitive since if we expand $h$ in a power series at $z_0$, $h(z)=\sum_{n=0}^\infty a_n(z-z_0)^n$, then $$|w-z_0||h(w)| = |w-z_0|\sum_{n=0}^\infty a_n|w-z_0|^n &lt; |z-z_0|\sum_{n=0}^\infty a_n|z-z_0|^n.$$ for $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$. But this is of course not a proof. How do I set Titanfall in-game language to English (Russian version)? This question was initially about Russian PC version of Titanfall, available in Russian Origin store.  It appears that there exists a Worldwide PC version of Titanfall for which .  Tech support said that Russian users can only buy Russian version of Titanfall (both normal and Deluxe editions) and that they will not have any other in-game language available. They also offered me a refund, which I accepted.    I'm living in Russia and I recently pre-purchased Titanfall from here. My version doesn't have an option to select game language other than Russian.  Russian version has awful localization, many mistranslated technicalities and poorer game UI fonts than its English counterpart. Titan's on-board AI voice is pretty lame. The campaign briefing lines are cut off to fit the English files' length. Voice actors took liberty to change a few things to their liking. Audio effects like reverberation, echo and such are distinctively more basic. Overall audio quality is much lower, which greatly hinders immersion.  When I was playing open beta, it was all in English and considerably better. The game also doesn't support Russian characters so everyone is using transliteration in chat.  Switching Origin UI language to English didn't change the game's language. It's still all Russian.  How do I change Titanfall UI and audio to English?    Also: the support person didn't explicitly say tampering with localization files was against any rules or policy, so this may or may not be okay to do. Gotta find some hard info on that.  Although, adding launch parameters -language english, -language "english" in case of Russian version does not help. Players with other versions report that it does help.  As can be seen during game installation, the sound files alone take up about ~33 GB of disk space, and according to Origin's solution to Worldwide PC version, these files contain localization for every language available. It's just the rights to access them depend on version.    Note: it's important that the solution does not break any rules and is not outright illegal. Otherwise it may be removed from this site by a moderator, or, in worst case scenario, lead to some player's game and account being locked. So if you post something suspicious looking, be sure to include some links to official statements that allow such actions.