How can I install Ubuntu on my Lenovo Yoga 500 Laptop? Is there a better way than a virtual machine? I have to use Ubuntu for my work.
I would like to see a full how-to guide on how to install Ubuntu.
Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them.   Let $a$ be the primitive root then I know other primitive roots will be among $\{a,a^2,a^3 \cdots\cdots a^{\phi(n)} \}$ because any other number will be congruent modulo $n$ to one of these. Then I figured as the answer is $\phi(\phi(n))$ so for $a^k$ to be primitive root, $k$ must be coprime with $\phi(n)$.But I don't know the reason.Am I missing any silly point? If $a_0\in R$ is a unit, then  $\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$ Windows service stops automatically String manipulation with & or + in VB.NET Does a constantly accelerating charged particle emit EM radiation or not? The  gives the recoil force, $\mathbf{F_{rad}}$, back on a charged particle $q$ when it emits electromagnetic radiation. It is given by:  $$\mathbf{F_{rad}} = \frac{q^2}{6\pi \epsilon_0 c^3}\mathbf{\dot{a}},$$  where $\mathbf{\dot{a}}$ is the rate of change of acceleration.  If a particle has a constant acceleration, $\mathbf{\dot{a}}=0$, then there is no reaction force acting on it. Therefore the particle does not lose energy.   Does this mean that a constantly accelerating charged particle does not emit electromagnetic radiation? How to store credit card info on iphone? I quite often add a question as a favourite when there is something of interest to me in it or its answers. Looking back through my favourites, I cannot remember why I added some of them as a favourite.  What I'd like to be able to do is to add a note/comment when I add a favourite as a reminder as to why I've added it. This will show on my favourites page, but nowhere else - and possibly to no one else.  In addition, it would be good to be able to favourite an answer to a question, thereby adding a favourite to both the question and a particular answer with something of note in it. This would only show as a single favourite, but when clicking its link would go to the favourited answer.  Possible? Or is there another way of doing this? I've already got too many favourites to remember why I added a lot of them. Basically this suggestion for the Android app:   Many times I'll see that a post has been edited on the Android app. I'll want to see exactly what was edited, and the Android app doesn't show this. Can the revision history be added to the Android app? INI file parsing in PowerShell I am a girl in my early twenties traveling to USA from India and in my visa application I have provided the contact address for a relative in San Francisco, and their invitation letter. I would like to visit LA first. Will I face problems during immigration if I land in LA (with an immediate on wards ticket to SF)? I dont plan to get on the flight from LA to SFO.  Initially I was planning to fly to SFO from India and then fly to LA immediately after completing immigration at SFO airport. How to archive the whole website? To save a single page via Wayback Machine I can go to:      How can I archive the website recursively by Wayback Machine?    There is some project such as , but I'm looking for a feature which allows me the upload the website recursively. Is there a way to make PythonTEX compile as of today in Overleaf V2 ?  \documentclass[]{article} \usepackage[utf8]{inputenc} \usepackage{pythontex}  \begin{document} \begin{pyverbatim} def calc_ellipticity(theta, delta):     theta = np.deg2rad(theta)     psi = np.arcsin(np.sin(2*theta) * np.sin(delta))/2     return np.tan(psi) \end{pyverbatim} \end{document}   Compiling gives : ? PythonTEX ? What can be said about the convergence on $|z|=1$? I already found the radius of convergence of the power series $:$ $$\sum_{n=1}^{\infty} \frac {(-1)^n} {n} z^{n(n+1)}.$$ What happens at the boundary of disk of convergence?  I found that the radius of convergence to be $1$ and hence the disk of convergence is $|z|&lt;1$. Now I found at $z=1$ the series converges. Now how can I proceed? Please help me. Thank you in advance. No cycle containing edges $e$ and $g$ implies there is a vertex $u$ so that every path sharing one end with $e$ and another with $g$ contains $u$ There is a proof in my textbook for the following claim, which doesn't make a whole lot of sense to me.  My annotations are in bold.  Could someone perhaps elaborate on what's going on?  Claim.  If there does not exist a cycle containing edges $e$ and $g$ then there exists a vertex $u \in V (G)$ such that every path in $G$ sharing one end with e and another with $g$ contains $u$.  Proof: The claim trivially holds if $e$ or $g$ is a loop OK, so we assume that neither is. Let $P$ with vertex set $v_1, v_2, . . . , v_k,$ in order, be a path with $e$ joining $v_1$ to $v_2$ and $g$ joining $v_{k−1}$ and $v_k$. Let $f_i \in E(P_i)$ be the edge with ends $v_i$ and $v_{i+1}$. Let $j$ be chosen minimum so that no cycle in $G$ contains $e$ and $f_j$ We can do this because we know that at least the edge $g$ will not create a cycle by assumption, right?. We will show that $u = v_j$ satisfies the claim.  Suppose not. Let $C$ be a cycle containing $e$ and $f_{j−1}$ What if $f_j = e$?  then what cycle? a single vertex? and let $P′$ be a path from an end of $e$ to an end of $f$ Not sure what the book meant by $f$ here, any guesses? avoiding $u$. Choose a subpath $Q$ of $P′$ with one end in $V (C)$ and another in ${v_{j+1}, v_{j+2}, . . . , v_k}$ as short as possible. Then $C \cup Q \cup P$ contains a cycle containing both $e$ and $f_j$, a contradiction. (The last statement requires some case checking.) This ending seems a bit abrupt and non-obvious to me  Thanks for the help Find the remainder of $\dfrac{2^{2014}}{7}$   I am new to the modular arithmetic, Any suggestions to solve this question? Simulating group_concat MySQL function in Microsoft SQL Server 2005? Why would an incorrect password attempt take a lot longer to process than a correct one? The most prominent place I've noticed this is when SSH-ing at work, but I feel like I've observed this behaviour elsewhere too.  When I try to log into Linux servers from my Windows desktop at work, I've noticed that if I mis-type my password, it takes a good 5 seconds or so before I get "Access Denied" back. Then when I type my password correctly, the login (along with welcome messages etc) is virtually instant.  Is there any logical reason for this, or would it be down to some odd configuration that's specific to the machines here at work? $127$ has an interesting property: It is the $31$st prime number and its rank ($31$) is also a prime. $31$ is the $11$th prime so its rank is also a prime. $11$ is also a prime number with a rank ($5$) that is also a prime. $5$ is the 3rd prime number and and so its rank ($3$) is also a prime. And finally $3$ is the $2$nd prime so its rank is also a prime... Is there a name for primes whose rank (index in the prime series) is also a prime? That is:     If $a_i$ is the $i$th prime number, then $i$ is a also a prime.   How about numbers like $127$ where going down ranks of ranks always produce prime numbers (down to rank $2$ of course)? The first $11$ :-) primes in this (infinite) series would be:  $3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, \dotsc$  Many thanks! How to sample from a discrete distribution? Assume I have a distribution governing the possible outcome from a single random variable X.  This is something like [0.1, 0.4, 0.2, 0.3] for X being a value of either 1, 2, 3, 4.  Is it possible to sample from this distribution, i.e. generate pseudo random numbers upon each of the possible outcomes given the probability of that outcome. So if I wanted to know what the probability of getting a 2 is, the sampling operation may return 0.34 or something like that.  The reason I ask is that I'm trying to implement an action selection policy for a reinforcement learning method based on a research paper. From what I gather from the paper, the author is able to sample the distribution by "mapping the uniform distribution U[0,1] through cumulative probability density functions obtained by adaptive numerical integration". From this he then samples the transition probabilities for each trial...  I would be grateful for any info on this...  Thanks in advance How do you make corrections to a published paper?