This is written in the beginning of "LOTR: Fellowship of The Ring" novel.   Tolkien invented many languages. But It doesn't match with any languages I have seen on internet. Can anyone tell me which language of Tolkien is it? Also can anybody give me a link to all of Tolkien's languages?
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.
How to prevent the paragraphs to spread out over the page? I have a problem. I have several large equations within the text. And when LaTeX does not manage to put the equation on the same page and moves it to the next one, he spreads the paragraphs of the text in the first page. How could I prevent this? I know about \pagebreak and \newpage, but I do not want to force the \newpage, as the text may move (if I add something later or so on) and the pagebreaks may not be valid. I would like to make LaTeX not to spread the paragraphs over a page automatically. Is it possible?  I tried to make an example:  \documentclass[a4paper,12pt,twoside]{book} \usepackage{amsmath} \begin{document}  LaLaLa\\\  LaLaLa\\\  LaLaLa\\\  LaLaLa\\\  \begin{eqnarray} 123&amp;+&amp;123 \\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ \end{eqnarray}  LaLaLa\\\  LaLaLa\\\  \begin{eqnarray} 123&amp;+&amp;123 \\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ \end{eqnarray}  \end{document} I'm taking Calculus II next semester so I don't have all the tools from it to back me up but something came up in a discussion this evening that asked me to prove that the following sequence converges. As I'm new to formal math I'm hoping someone can verify my proof, especially step $6$. I must prove that $$\lim_{n\to\infty}P_n=b$$ where $$P_{n+1} = P_n + \frac{b-P_n}{2}$$ $$P_0 = a$$ and $a,b \in \mathbb{R}$ such that $a&gt;b$ My proof goes as follows:   We must prove that $$\lim_{n\to\infty}P_n=b$$  That is, $\forall\epsilon&gt;0$ there exists a $\delta$ such that there exists an $n$ satisfying $$n&gt;\delta \iff b-P_n&lt;\epsilon$$  First, let us prove that $P_n$ is bounded by $b$ by contradiction. Let us choose an $i$ such that $P_i &lt; b$ and $P_{i+1}&gt;b$. Then we have $P_{i+1} = P_i + \Delta P$ where $\Delta P &gt; b-P_i$. but this is impossible because by the definition of the recurrence $\Delta P = \frac{b-P_i}{2} &lt; b-P_i$. Therefore, $P_i$ is bounded from above by $b$.  Additionally, it is trivial to show that $P_{i+1} &gt; P_i$.  With this in mind, choose an arbitrary $\epsilon$. We need to show that there exists an $n$ such that $P_n &gt; b-\epsilon$. Assume the contrary, that $P_n$ is bounded by $\epsilon$.  Using a similar argument to what which was used before, assume $P_n$ is bounded by $\epsilon$ and then there exists a $P_i$ such that $P_{i+1}\not &gt; P_i$. But this results in a contradiction with (4)  Therefore, $P_n$ is not bounded by $\epsilon$ and we have proved what we want to prove. How can slightly blurred photos be improved in post processing? Is it normal for focal length to change while focusing? When I change the focus distance on my Pentax 55-300mm lens, the focal length appears to change.  More specifically, the lens appears to zoom in as I focus towards the minimum focusing distance.  This is most noticeable at the 55mm end of the lens.  Is this normal? Finding the limit of roots products $(\sqrt{2}-\sqrt[3]{2})(\sqrt{2}-\sqrt[4]{2})(\sqrt{2}-\sqrt[5]{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt[n]{2})$ What can I do when getting "We are no longer accepting questions/answers from this account"? If you are reading this, you may have been affected by one of our post quality bans. If so, it is important that you  to understand what has happened and how to regain the permissions that you lost. While trying to ask a question, one could see:  We are no longer accepting questions from this account. See  to learn more.  Likewise, for answers:  We are no longer accepting answers from this account. See  to learn more.   Why am I getting this message? Are deleted posts taken into account too? Is a question/answer ban the same as a suspension? How do I avoid getting a question ban? How long do I have to wait before I can post again? What can I do to release the ban? How can I reactivate my account? Can I simply create a new account? I'll just ask somewhere else on the SE network, and they'll migrate my question to the correct site! Will a ban on one Stack Exchange site affect my standing on other sites in the network? Does this apply to meta sites too? My account is in good standing. Why am I still blocked? How do I set a featured image (thumbnail) by image URL when using wp_insert_post()? While looking through the , I noticed that there's no parameter in the array it requires which will allow me to set the 'Featured Image' for a post, displayed as the  in my theme.   I have looked into functions like , as suggested by Mr. Bennett, but this seems to be a relatively new addition to WordPress itself and the WordPress codex. As such, there aren't any sources that I can find which explain how the $thumbnail_id parameter should be acquired and supplied. If this really is the function to use, in what way could I provide it with a valid $thumbnail_id parameter when all I have is an image URL?  Thanks in advance! How to burn a Windows .iso to a USB device? How can I determine whether my processor is 32-bit or 64-bit? How can I determine whether my processor is 32-bit or 64-bit? My processor is AMD Phenom 8450 triple-core. Is there a size of rectangle that retains its ratio when it's folded in half? A hypothetical (and maybe practical) question has been nagging at me.  If you had a piece of paper with dimensions 4 and 3 (4:3), folding it in half along the long side (once) would result in 2 inches and 3 inches (2:3), which wouldn't retain its ratio. For example, here is a piece of paper that doesn't retain its ratio when folded:   Is retaining the ratio technically possible? If so, what is the side length and ratio that fulfills this requirement? Any help would be appreciated.  Update:  I added "once" because I got an answer saying that any recectangle would work, as any rectangle folded twice has the original ratio. Nice answer, but not quite what I was looking for. As for the other answers, I got 3x as much information as I needed! Thanks! How to remove duplicate values from an array in PHP How can I remove duplicate values from an array in PHP? What is the difference between procfs and sysfs? What is the difference between procfs and sysfs?  Why are they made as file systems? As I understand it, proc is just something to store the immediate info regarding the processes running in the system. Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$ Storage for Eldritch Horror Is there an optimal strategy for storing Eldritch Horror cards and tokens so that game play is facilitated.   I am not looking for answers that including "zip bags" and other things to keep the game element safer while is storage. Although, a strategy that works well for both cases will clearly be superior to one that only works for game play.  I am not limiting this to commercial solutions. I am quiet happy to build your custom made solution if you release the specification/plans of course!  Although, it it requires a 3D printer, more power to you. Currently, the only way to ask for something to be migrated to a site that isn't part of the standard migration path is through a custom mod note.  I currently have two active custom mod notes on Stack Overflow that ask for migration that are not standard targets for SO.  I believe the questions could be better answered on another site.  Fortunately, or unfortunately, the questions have been answered, and answers have been up voted and accepted. At this point, a week and a half later, migrating either of the questions would be disruptive to both communities and not necessarily produce a better answer (they've got accepted answers).  The unfortunate part is I really do believe that both questions could have gotten better answers and more attention on another site.  What I would like to see is a more streamlined way to flag for moderator attention for migrations.  The UI would look something like this:    This would allow people to find the proper site easily.  No more "Migrate this to Mathamatica.SE" because, well, that doesn't exist.  Something that I didn't do on this mock up is add a link to the corresponding help/on-topic page for the target site.  Something people could check and click before hitting the submit button.  I don't have this quite right again, because while that shows a "closing for off topic", this is a flag and not a vote to close.    The "flag for migration" is only one part of this though.  The other part would be to make another flag queue style thingy for moderators in which they could look at just migration flags created through the above interface.  By providing the mods with a "just migration" view into the flags, questions that should be migrated can be quickly identified from the multitude of existing flags and handled promptly before the migration would be disruptive. StandardSetController with History records Sum of indepedent random variables and a constant Let $X_1$ and $X_2$ be independent Normal random variables with mean $\mu_1$ and $\mu_2$, and variances $\sigma_1$ and $\sigma_2$. Let $Y = X_2-X_1 + c$, where $c$ is a constant. For notational simplicity, let $Y = X' + c$, where $X' = X_2-X_1$ The moment generating function of a Normal random variable $X_n\sim N(μ_n,σ_n^2)$ is $$M_{X_n } (t)=\text{exp}\Big(μ_n t+\frac{σ_n^2 t^2}{2}\Big)$$ Then, $$M_{X'} (t)=\text{exp}\Big[t(\mu_1+\mu_2)+\frac{t^2}{2}(\sigma^2_1+\sigma^2_2)\Big]$$ Thus, by the uniqueness property of moment generating function, $X'$ is said to follow a Normal distribution with mean $(\mu_1+\mu_2)$ and variance $(\sigma^2_1+\sigma^2_2)$. How do I proceed from here to derive the following:  First, I want to establish that $Y$ is also a Normal random variable using the moment generating function. The constant $c$ bugs me, but I cannot ignore it. As per my understanding, $Y\sim N\big(c+\mu_1+\mu_2,\sigma^2_1+\sigma^2_2\big)$ (and I could be completely wrong). But I don't understand how to apply the moment generating function to prove that.  Secondly, I want to derive $P(Y \leq a)$, where $a$ is a constant. If I am not mistaken, $P(Y \leq a)$ can be written as $$P(Y \leq a)=\Phi\Bigg(\frac{a-X_2-X_1 + c}{\sqrt{2(\sigma^2_1+\sigma^2_2)}}\Bigg).$$ But how do I arrive at that.   Any help is high solicited and appreciated. Two paths that show that $\frac{x-y}{x^2 + y^2}$ has no limit when $(x,y) \rightarrow (0,0)$ style=apa causes bibliography to fail Is it possible to programmatically stop and start ArcGIS Server map services using Python?