The Dwarf Rings are being mentioned at the start of LOTR but no mention is made of them elsewhere despite the nine and three gathering heaps of attention.  While the dwarfs of ME are renowned for sticking to their mining and delving and smithing, only rarely coming out for light and (let's face it) the Elves aren't exactly the friendliest bunch, surely some news of the Dwarf Rings would have come to the attention of the Free peoples?  So what happened to the seven
In the Lord of the Rings it is said     "In Eregion long ago many Elven-rings were made, magic rings as you   call them, and they were, course, of various kinds: some more potent   and some less. The lesser rings were only essays in the craft before   it was full-grown, and to the Elven-smiths they were but trifles — yet   still to my mind dangerous for mortals. But the Great Rings, the Rings   of Power, they were perilous.”   [LotR I 2 (60)]   There is a clear distinction here between the 20 Rings of Power in the Ring Rhyme (the 7,9,3,and 1) and these other rings. What happened to the "lesser rings"?
What is a word that means "someone who pretends to be your friend but is actually your enemy?" How do I boot into single-user mode from GRUB? Prove that the minimum number of cycles is $m-n+1$ The question I have is:   Prove that the minimum number of cycles is $m-n+1$ in a connected graph. Where one cycle is a path that starts that begins and ends at the same vertex. Where $m$ is edges and $n$ is the vertices.   I have no idea where to start. A few hints would be appreciated. Please do not provide me with the answer. Why is spawning threads in Java EE container discouraged? One of the first things I've learned about Java EE development is that I shouldn't spawn my own threads inside a Java EE container. But when I come to think about it, I don't know the reason.  Can you clearly explain why it is discouraged?   I am sure most enterprise applications need some kind of asynchronous jobs like mail daemons, idle sessions, cleanup jobs etc.  So, if indeed one shouldn't spawn threads, what is the correct way to do it when needed? Why do some AC adapters and power supplies generate a whining noise, and what can I do about it? Meta duplicates don't require an answer. Don't suggest there is one when that's not true While learning about Random variables I came across the mean of random variable X.  The definition says that the expected value of random variable E(X) = Mean of Random variable X  I am not able to understand why is that so. Can any one please help me with it How would I have a  action that may have some effects on the current page but would also change the URL in the browser so if the user hits reload or bookmark, then the new URL is used?    It would also be nice if the back button would reload the original URL.  I am trying to record JavaScript state in the URL. Could 'cogito ergo sum' possibly be false? In unix/linux, any number of consecutive forwardslashes in a path is generally equivalent to a single forwardslash. eg.  $ cd /home/shum $ pwd /home/shum $ cd /home//shum $ pwd /home/shum $ cd /home///shum $ pwd /home/shum   Yet for some reason two forwardslashes at the beginning of an absolute path is treated specially. eg.  $ cd ////home $ pwd /home $ cd /// $ pwd / $ cd // $ pwd // $ cd home//shum $ pwd //home/shum   Any other number of consecutive forwardslashes anywhere else in a patch gets truncated, but two at the beginning will remain, even if you then navigate around the filesystem relative to it.  Why is this? Is there any difference between /... and //... ? How to calculate hash value of a file in Java? If someone is my proxy, I am their _____? Bonus question: If I am a proxy for multiple people, what's the collective noun for that group? is it possible to create this kind of material with the standard textures? The Voronoi texture (set to cells) is coming close but I don’t know if it is possible to a) set the single cells apart from each other and b) to give them seperate colors. When I try to create a reaction from chemmacros/reactions I got the following error. I dunno what I am missing. Please help. Thank you.     line 11: Undefined control sequence. \end   line 11: Missing number, treated as zero. \end   line 11: Missing number, treated as zero. \end     Minimum Working Example    \documentclass[12pt,a4paper]{article} \usepackage{chemmacros} \chemsetup{modules={reactions}} \begin{document} \begin{reaction}     A + B -&gt; C \end{reaction} \end{document} I need some advice as I am struggling with the following combinatorics exercise.  Let $k$ be a given positive integer. Show that any non-negative integer $N$ can be written uniquely in the form $$N=\binom{x_k}{k}+\binom{x_{k-1}}{k-1}+\dots+\binom{x_1}{1},$$ where $0 \leq x_1 &lt; \dots &lt; x_{k-1} &lt; x_k$.  The exercise is accompanied by this advice:  Let $x$ be such that $\binom{x}{k} \leq N &lt; \binom{x+1}{k}$. Then any possible representation has $x_k = x$. Now use induction and the fact that $N-\binom{x}{k} &lt; \binom{x}{k-1}$ to show the existence and uniqueness of the representation.  I am really stuck and thus grateful for the slightest hint on how to solve it. Question about repeating decimal? For simple fraction, we can easily convert it to repeating decimal by calculator. Ex. $\frac 1 3 = 0.33333\ldots$, $\frac 1 7=0.(142857),\ldots$ But some fraction fraction like $10/29, 1/97,...$ The repeating part of them are too long, so it can't fully show on the calculator. So is there any algorithm  to find the repeating part for that fraction? Is it possible to install Python 3.x in 14.04 LTS, or upgrade from Python 2.7.6 to 2.7.8? Is there a max array length limit in C++? Is there a max length for an array in C++?  Is it a C++ limit or does it depend on my machine? Is it tweakable? Does it depend on the type the array is made of?  Can I break that limit somehow or do I have to search for a better way of storing information? And what should be the simplest way?  What I have to do is storing long long int on an array, I'm working in a Linux environment. My question is: what do I have to do if I need to store an array of N long long integers with N > 10 digits?  I need this because I'm writing some cryptographic algorithm (as for example the p-Pollard) for school, and hit this wall of integers and length of arrays representation. It is quite easy to show that for every prime $p$ and $0&lt;i&lt;p$ we have that $p$ divides the binomial coefficient $\large p\choose i$; one simply notes that in $\large \frac{p!}{i!(p-i)!}$ the numerator is divisible by $p$ whereas the denominator is not (since it is a product of numbers smaller than $p$ and $p$ is prime).  My problem is with generalizing this argument for $q=p^n$. I'm looking for the most elegant and simple way to prove that $p$ still divides $\large q\choose i$. How does a  work, and what differentiates it from other linear classifiers, such as the , , or ? *  (* I'm thinking in terms of the underlying motivations for the algorithm, optimisation strategies, generalisation capabilities, and run-time complexity)