Prove $(1+\sqrt{2})^k= \sqrt{n}+\sqrt{n-1}$
Expansion of $(1+\sqrt{2})^n$
When should I use "shall" versus "will"? Which is the correct use of these two words, and in which context should one be used rather than the other? note: This is a question about relativistic effects. I've included some detail about the spacecraft and its orbit for background, but the question is about relativistic effects and their observability.     When the Parker Solar Probe passes within 8.5 solar radii of the Sun, it will be moving really fast.  A perihelion and aphelion of about 6.6 and 109 million kilometers suggests a semi-major axis of 57.8 million kilometers. The standard gravitational parameter for the Sun is:  $$GM_{Sun} \ = 1.327\times 10^{20} m^3/s^2$$  So using the :  $$v^2 \ = \ GM_{Sun}\left(\frac{2}{r}-\frac{1}{a}\right)$$  the orbital velocity at perihelion will be about 195,000 m/s, or about 25 times faster than a satellite in LEO. This is close to one part per thousand of the speed of light! I think this will be a new record for a spacecraft velocity relative to the solar system.  Depth of a gravitational potential well for an object of mass $m$ is approximately given by:  $$u_G \ = \ -GM_{Sun}\frac{1}{r}$$  At 6.6 million kilometers the spacecraft will be 23 times deeper in the Sun's gravitational well than it would be at 1 AU.  Will there be any particularly unique relativistic effects detectable for the Parker Solar Probe during it's close flyby of the Sun?  Just for example, during one close flyby, how much time will its on-board clock gain or lose compared to some convenient standard solar system timescale? I know there are several different choices, "GPS time" might be one but I don't want to arbitrarily choose an inconvenient one.     above: Parker Solar Probe cropped, from .    above: screen shot from NASA's      note: the link seems to be dead now, but there is plenty of related information available at     above: Illustration of the Parker Solar Probe's orbit, from . If it was possible to dig a hole that went from one side of the Earth to the other... ...And you jumped in.  What would happen when you got to the middle of the Earth? Would you gradually slow down, until you got to the middle and once you were in middle would every direction feel like it was up? Why does strcmp() return 0 when its inputs are equal? When I make a call to the C string compare function like this:  strcmp("time","time")  It returns 0, which implies that the strings are not equal.  Can anyone tell me why C implementations seem to do this? I would think it would return a non-zero value if equal. I am curious to the reasons I am seeing this behavior. Find all $x$ such that $$x^6=(x+1)^6.$$   So far, I have found the real solution $x= -\frac{1}{2}$, and the complex solution $x = -\sqrt[3]{-1}$.  Are there more, and if so what/how would be the most efficient find all the solutions to this problem? I am struggling to find the rest of the solutions. Give $a, b. c$ be the lengths of a triangle and $a+ b+ c= 3$. Prove that: $$\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq  \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$$ My try  We have England MO inequality: $$\left ( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a} \right )^{2}\geq  \left ( a+ b+ c \right )\left ( \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c} \right )$$  and $$\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a} \geq  3=  a+ b+ c$$  But it cannot be used to prove the above inequality and how to use triangle inequality in this inequality. I need the help. Thanks! Prevent iTunes from starting when iPhone/iPod/iPad is plugged in on Windows Is there any way to prevent iTunes from starting under XP when an iPhone is plugged in?  Using iTunes v 9.  Usually I just want to charge my phone and don't want iTunes to start.  Note: Exact  of this question - except this is for v9 on Windows.  The answer in the related question doesn't work for me. maximize volume of triangle in given perimeter Difference in meaning and use of future perfect forms Also how do LEFT JOIN, RIGHT JOIN and FULL JOIN fit in? Can I re-create deleted Gmail accounts? A long time ago, I created a Google mail account. I fell for a trap that was common back then (not sure how it is nowadays) and ended up with two Google accounts—one for Gmail (using the Gmail email address I would like to use) and one for everything else (using an email address at an external mail provider).  Hoping that I could associate my Gmail and my actual Google account, I deleted my Gmail account only to discover that deleted account names cannot be used again. So basically my old Gmail account along with the Google email address I would like to use is gone.  My main Google account got assigned a different Gmail account in the mean time.  Do you know if it is possible to re-use my preferred Gmail address? If possible, I would like to assign it to my main Google account. But I'd also be happy to have it as a second Google account, anything is better than the current situation. I have a script which calls two commands:  long_running_command | print_progress   The long_running_command prints progress but I'm unhappy with it. I'm using print_progress to make it nicer (namely, I print the progress in a single line).  The problem: Connection a pipe to stdout also activates a 4K buffer, so the nice print program gets nothing ... nothing ... nothing ... a whole lot ... :)  How can I disable the 4K buffer for the long_running_command (no, I do not have the source)? Sylow Theorem Explanation The Sylow theorems say that all $p$-Sylow subgroups are conjugate.   Is that the only thing they are conjugate to? In other words could a $p$-Sylow subgroup also be conjugate to something that is not a $p$-Sylow group?   Or does the fact that every element with order dividing $p$ is in a $p$-Sylow group mean that that is never possible? Combine and Minify Multiple CSS / JS Files I have seen lots of UI designer nowadays including this kind of illustration i want to know the name of this style so that i can also learn and try this in my designs. The question is about the encryption equation in asymmetric encryption.  For $c\equiv m^e\bmod n$, prove that for every unique $m$ there is a unique $c$.  Here $n=pq$ where $p$ and $q$ are primes larger than $m$ and the exponent $e$, is $\geq3$ and coprime with $p-1$ and $q-1$.  Thanks! I have been attempting this problem for a while, it is an assignment problem so I don't want somebody to just post the answer, I'm just looking for hints.  Let $\mathfrak{u}(n,\mathbb{C})$ be the Lie algebra of strictly upper triangular $n\times n$ matrices over $\mathbb{C}$.  I am trying to show that $\mathfrak{u}(n,\mathbb{C})$ is a nilpotent Lie algebra for $n\geq 2$.  This is what I have looked at so far.  I have done some explicit calculations (I shall omit the bulk of them and just summarise below) for $n=2,3,4$ and have come up with some ideas.  Let $L=\mathfrak{u}(n, \mathbb{C})$.  If $n=2$,  $L^1 = [L, L] = &lt;0&gt;_{\mathbb{C}}$  So $L$ is nilpotent.  If $n=3$,  $L^1 = [L, L] = &lt;e_{13}&gt;_{\mathbb{C}}$  $L^2 = [L, L^1] = &lt;0&gt;_{\mathbb{C}} = \left\{ 0 \right\}$  So, $L$ is nilpotent.  If $n=3$  $L^1 = [L,L] = &lt;e_{13},e_{14},e_{24}&gt;_\mathbb{C}$  $L^2 = [L,L^1] = &lt;e_{14}&gt;_{\mathbb{C}}$  $L^3 = [L, L^2] = &lt;0&gt;_{\mathbb{C}} = \left\{ 0 \right\}$  So, $L$ is nilpotent.  So my hypothesis so far is that for an arbitrary $m\in\mathbb{N}$, $L^{m-1}=\{0\}$  My instinct is to do this by induction. My base case, $n=2$ has already been shown. Now I assume that $\mathfrak{u}(n,\mathbb{C})$ is nilpotent for $n=k$ ($k\in\mathbb{N}$) and consider the case for n=k+1.  However, I have no idea how to link these two cases together. I have read in Introduction to Lie Algebras and Representation Theory - J.E. Humphreys that $L^k$ should be spanned by basis vectors $e_{ij}$ where $j-i=k+1$ - however in my calculations for $L^1$, I obtain an $e_{14}$ which does not satisfy 4-1=2.  Am I missing something completely obvious? Or have I made a mistake somewhere? (I have checked my calculations numerous times, but cannot find any errors!)  Thanks,  Andy. I'm getting an error in WSL while trying to use .  I followed all the install steps successfully on Windows 10 x64 using the steps at , then Ubuntu for Windows with GitHub desktop .  But I'm blocked at the latest steps.  It's where I trying to download or train the model in Windows. Using for example the Ubuntu Linux prompt, I navigate manually to the appropriate place and type a following command like the one below (I also tried in su mode)  bash pretrained_models/download_pix2pix_model.sh facades_label2photo   I get an immediate error, always the same, similar to these:  root@Azure:/mnt/c/Users/vincent/Downloads/vision/pytorch-CycleGAN-and-pix2pix# bash pretrained_models/download_pix2pix_model.sh facades_label2photo pretrained_models/download_pix2pix_model.sh: line 2: $'\r': command not found Note: available models are edges2shoes, sat2map, and facades_label2photo pretrained_models/download_pix2pix_model.sh: line 4: $'\r': command not found ]pecified [facades_label2photo pretrained_models/download_pix2pix_model.sh: line 6: $'\r': command not found mkdir: cannot create directory ‘./checkpoints/facades_label2photo\r_pretrained\r’: No such file or directory pretrained_models/download_pix2pix_model.sh: line 10: $'\r': command not found WARNING: timestamping does nothing in combination with -O. See the manual for details.  : No such file or directoryhphoto pretrained_models/download_pix2pix_model.sh: line 12: $'\r': command not found pretrained_models/download_pix2pix_model.sh: line 13: $'\r': command not found   Any idea? Prove that$$\overbrace{\binom n 0 + \binom n 1 + \binom n 2 + \cdots + \binom{n}{\frac{n-1} 2}} + \overbrace{\binom n {\frac{n+1} 2} + \cdots + \binom n {n-1} + \binom n n} = 2^n. $$   Background:  This problem came from the following link:    and I didn't understand why that was true in which no one on the page has proved it to be so. and i was wondering what was the proof on that I have a webpage that implements a set of tabs each showing different content. The tab clicks do not refresh the page but hide/unhide contents at the client side.  Now there is a requirement to change the page title according to the tab selected on the page ( for SEO reasons ). Is this possible? Can someone suggest a solution to dynamically alter the page title via javascript without reloading the page?