Interpretation of the t.test in R
Interpretation of the t.test and Wilcox.test in R
White space below footer Are the complements of two homeomorphic compact, connected subsets of $\mathbb{R}^2$ homeomorphic? Finding derivative of $\sqrt[3]{x}$ using only limits I need to finding derivative of $\sqrt[3]{x}$ using only limits  So following tip from : I multiplied top and bottom by conjugate of numerator  $$\lim_{h \to 0} \frac{\sqrt[3]{(x+h)} - \sqrt[3]{x}}{h} \cdot \frac{\sqrt[3]{(x+h)^2} + \sqrt[3]{x^2}}{\sqrt[3]{(x+h)^2} + \sqrt[3]{x^2}}$$  $$= \lim_{h \to 0} \frac{x+h-x}{h(\sqrt[3]{(x+h)^2} + \sqrt[3]{x^2})}$$  $$= \lim_{h \to 0} \frac{1}{\sqrt[3]{(x+h)^2} + \sqrt[3]{x^2}}$$  $$= \frac{1}{\sqrt[3]{x^2} + \sqrt[3]{x^2}}$$  $$= \frac{1}{2 \sqrt[3]{x^2}}$$  But I think it should be $\frac{1}{3 \sqrt[3]{x^2}}$ (3 instead of 2 in denominator?)  UPDATE  I found that I am using the wrong conjugate in step 1. But this (wrong) conjugate gives the same result when I multiply the numerator by it. So whats wrong with it? (I know its wrong, but why?) How can I model flips until N successes? You and I decide to play a game where we take turns flipping a coin. The first player to flip 10 heads in total wins the game. Naturally, there is an argument about who should go first.  Simulations of this game show that the player to flips first wins 6% more than the player who flips second (the first player wins approx 53% of the time).  I'm interested in modelling this analytically.  This isn't a binomial random variable, as there are no fixed number of trials (flip until someone gets 10 heads). How can I model this?  Is it the negative binomial distribution?    So as to be able to recreate my results, here is my python code:  import numpy as np from numba import jit   @jit def sim(N):      P1_wins = 0     P2_wins = 0      for i in range(N):          P1_heads = 0         P2_heads = 0         while True:              P1_heads += np.random.randint(0,2)              if P1_heads == 10:                 P1_wins+=1                 break              P2_heads+= np.random.randint(0,2)             if P2_heads==10:                 P2_wins+=1                 break     return P1_wins/N, P2_wins/N   a,b = sim(1000000) Is there a continuous bijection from $[0,1]$ onto $[0,1] \times [0,1]$? That is with $I=[0,1]$ and $S=[0,1] \times [0,1]$, is there a continuous bijection $$ f: I \to S? $$  I know there is a continuous bijection $g:C \to I$ from the Cantor set $C$ to $[0,1]$. The square $S$ is compact so there is a continuous function  $$ h: C \to S. $$ But this leads nowhere. Is there a way to construct such an $f$?  I ask because I have a continuous functional $F:S \to \mathbb R$. For numerical reason, I would like to convert it into the functional $$ G: I \to \mathbb R, \\ G = F \circ f , $$ so that $G$ is continuous. What is the difference between the two "worlds" when going thru the day/night wormhole at the altar? Some websites and textbooks refer to $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ as the correct formula for the uncertainty principle whereas other sources use the formula $$\Delta x \Delta p \geq \hbar.$$ Question: Which one is correct and why?  The latter is used in the textbook "Physics II for Dummies" (German edition) for several examples and the author also derives that formula so I assume that this is not a typing error.  This is the mentioned derivation:     $\sin \theta = \frac{\lambda}{\Delta y}$      assuming $\theta$ is small:      $\tan \theta = \frac{\lambda}{\Delta y}$      de Broglie equation:      $\lambda = \frac{h}{p_x}$      $\Rightarrow \tan \theta \approx \frac{h}{p_x \cdot \Delta y}$      but also:      $\tan \theta = \frac{\Delta p_y}{p_x}$      equalize $\tan \theta$:      $\frac{h}{p_x \cdot \Delta y} \approx \frac{\Delta p_y}{p_x}$      $\Rightarrow \frac{h}{\Delta y} \approx \Delta p_y \Rightarrow \Delta p_y \Delta y \approx h$      $\Rightarrow \Delta p_y \Delta y \geq \frac{h}{2 \pi}$      $\Rightarrow \Delta p \Delta x \geq \frac{h}{2 \pi}$   So: Which one is correct and why? if $a,b$ are nilpotent elements of a commutative ring $R$, show that $a+b$ is also nilpotent How can I determine the list of files in a directory from inside my C or C++ code?  I'm not allowed to execute the ls command and parse the results from within my program. How do I create a very long delay with Redstone? The only things I've found so far are a cart method (which ceased working in 1.6), a dispenser method (utilizing the decay on dropped items, but this requires occasional feeding), and a binary counter (which won't work for a single input but rather a fluctuating one).  I want to have a circuit that when I apply an input signal, the output signal doesn't change for 2 to 5 minutes.  When the input turns off, I again want a 2 to 5 minute delay.  I could accomplish this with (roughly) 240 to 600 redstone repeaters, but this is far in excess of being useful.  The only method I can think of, but would require a XOR gate (on the button-input and final output) to start and stop a clock attached to a binary counter.  While I can make room for a device of this size, I was hoping for something simpler. Definite integral, quotient of logarithm and polynomial: $I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$ I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{\lambda ^2x^2+\lambda x+1}\text{d}x =\frac{1}{\lambda}\int_0^{\infty}\frac{(\ln x-\ln \lambda)^2}{x^2+x+1}\text{d}x \\ =\frac{\ln ^2\lambda}{\lambda}\int_0^{\infty}\frac{1}{x^2+x+1}\text{d}x+\frac{1}{\lambda}\int_0^{\infty}\frac{\ln ^2x}{x^2+x+1}\text{d}x \\ =\frac{\ln ^2\lambda}{\lambda}\frac{2\pi}{3\sqrt{3}}+\frac{2}{\lambda}\int_0^1\frac{\ln ^2x}{x^2+x+1}\text{d}x$$ But I hav no idea on process the remaining integral:( Anyone knows how to solve it?  THX guys! I came with another method might work for this: Recall the handy GF $$\frac{1}{x^2+x+1}=\frac{2}{\sqrt{3}}\sum_{k=0}^{\infty}\sin \left(\frac{2\pi}{3}\left(k+1\right)\right)x^k$$ Then we have :  $$\int_0^1\frac{\ln ^2x}{x^2+x+1}\text{d}x=\frac{2}{\sqrt{3}}\sum_{k=0}^{\infty}\frac{2}{\left(k+1\right)^3}\sin \left(\frac{2\pi}{3}\left(k+1\right)\right)x^k$$ With $$\sum_{k=1}^{\infty}\frac{\sin (kx)}{k^3}=\frac{\pi ^2}{6}x-\frac{\pi}{4}x^2+\frac{x^3}{12}$$ What can we arrived? Book recommendations Any way to see mail sent by other user through shared address? My wife and I have a "shared" email address, like thesmiths@example.com which we both use from Gmail when we both want to get replies or for signing up for certain websites.  When an email is received at that address, it gets forwarded to both of our personal Gmail accounts.  The problem is that when either one of us "sends as" that address, the other does not see the sent message. So we have to cc or bcc each other (and she usually forgets.)  Is there any automagic way to have those sent emails appear in BOTH of our Gmail accounts?  I've tried filters but I don't think they work on outgoing messages. Basically we want a filter that would automatically forward emails from that address to the other person.  I know this can be done with standalone mail clients, but neither of us uses those. An inequality involving 4 real or complex numbers Proof that a Combination is an integer From its definition a combination $\binom{n}{k}$, is the number of distinct subsets of size $k$ from a set of $n$ elements.  This is clearly an integer, however I was curious as to why the expression  $$\frac{n!}{k!(n-k)!}$$ always evaluates to an integer.  So far I figured:  $n!$, is clearly divisible by $k!$, and $(n-k)!$, individually, but I could not seem to make the jump to proof that that $n!$ is divisible by their product. Limit of $\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} }$ How do I invoke a private static method using reflection (Java)? Truncated trivariate normal - conditional expectation Head or tail sound fine to my ESL ears.  What's the reasoning behind the plural usage?  I looked it up on etymonline but didn't find anything interesting. How to get only time from date-time C#