First off, I know that 0.9… = 1 and I'm not trying to prove or debate that, but my discussion about it is necessary for understanding the question. I was talking to my brother about whether 0.9… equaled one and I was using the “No holes in the number line proof” to get my point across when he said that 0.9… + (0.0…1)/2 would fit in between 0.9… and 1. Now I didn't know whether 0.0…1 was a valid number or not but I guess technically had to (if it's not feel free to tell me but for the rest of this we'll be pretending it is, think of it as a fun what if situation). I told him that it couldn't be divided because it's equal to zero, nothing could fit in between it and zero. He said (0.0…1)/2 could. I told him that assuming he could, it would equal 0.0…05 and infinity plus one equals infinity meaning that it could be simplified down to 0.0…5 and that 0.0…5 doesn't fit in between 0 and 0.0…1 meaning that 0.0…1 = 0 so 0.9… + 0.0…1 = 1  could be simplified down to 0.9… + 0 = 1, 0.9… = 1. He told me that infinity plus one does not equal infinity and I gave up. All that said and done, I was still left thinking about the numbers 0.0…1 through 0.0…9 and I realized they are quite weird. If we define x ≠ y as x having space in between it and y on the number line, and x = y as x having no space in between it and y on the number line  you will see where the weirdness starts. For just one example, if x = 0.0…1 and y = 0.0…2 then x = y. Now if z = 0.0…3 then y = z. But the thing is x ≠ z because y would fit in between them meaning that, y = x and y = z but x ≠ z. To keep this from breaking the transitive property of equality y would have to be both equal to and not equal to both x and z at the same time. That is why I have dubbed these numbers “Quantum Numbers”. Does anyone know whether or not anyone has published anything on them, I would be interested to know, and if you have anything to add please do.
I've just happened to read this  on MO (that of course has been closed) and some of the answers to a similar  on MSE.   I know almost nothing of nonstandard analysis and was asking myself if something like the sentence « $1- 0.999 \dots$ is a nonzero positive infinitesimal» could be easily expressed and proved in nonstandard analysis.  First of all, what is 0.999... ? If we take the usual definition as a series or as a limit of a sequence of rationals, then it will still be a real number and equal to $1$ (I guess by "transfer principle", but please correct me if I'm wrong).  Instead, let's define  $$0.9_N:=\sum_{i=1}^N 9\cdot 10^{-i} $$  where $N\in{}^*\mathbb{N}\setminus\mathbb{N}$ is an infinite nonstandard natural number. This $0.9_N$ is a legitimate element of ${}^*\mathbb{R}$, expressed as $0.$ followed by an infinite number of "$9$" digits.  What can be said about $\epsilon_N:=1-0.9_N$ ? Is there an elementary proof that $\epsilon_N$ is a positive infinitesimal of ${}^*\mathbb{R}$ ? (by "elementary" I mean just order and field axioms and the intuitive facts about infinitesimals, like that for $x$ infinite $1/x$ is infinitesimal etc.; no nonprincipal ultrafilters &amp; C).
How many integers between $1$ and $10^4$ contain exactly one $8$ and one $9$? I have this problem: How many integers between $1$ and $10^4$ contain exactly one $8$ and one $9$?, but i dont know if $18, 19, 28, 29...$ serve, else it could be $8$ y $9 = 2$ numbers, help please. Are there published predictions that hydrogen could remain metallic at ambient pressure? Below the question  there is a  that directed me to the BBC News summary . In their attempt to balance the skepticism of this particular reported observation with the potential implications of producing metallic hydrogen in a laboratory, the BBC included some far-reaching possibilities:  If that is true - and it is a controversial claim - it fulfills a more than 80-year quest to produce what many have said would be a wonder material. Theory suggests metallic hydrogen could be used to make zero-resistance electrical wiring and super-powerful rocket fuel, among many applications.  At the end of the article:  &quot;The scepticism here is probably a good thing, in that it will drive many groups towards attempting to reproduce this experiment. This publication will certainly incite the field. Again, if it holds up, this is an exciting result. I think in this case time will tell,&quot; he told BBC News.  ...referring to Marcus Knudson from Sandia National Laboratories.  And Jeffrey McMahon from Washington State University concurred: &quot;With respect to the tiny sample amount: Such experiments are performed in small diamond anvil cells. One challenge would be to make a larger quantity (at once); another, perhaps bigger challenge is to recover even the small sample (ie, remove it from the extreme pressures that it is under in the diamond anvil cell). &quot;Whether the latter is possible is an important open question.&quot;  Question: Are there published, peer-reviewed predictions that hydrogen could remain metallic at ambient pressure? A link or reference would be appreciated. If it's still in pre-print form (AeXiv) that's OK too. If it's possible, a little help understanding the prediction would be great as well: I know there are many predicted crystalline forms of water ice at elevated pressures and/or reduced temperatures, and many or most of these have been observed, but I think most of them revert once pressure is reduced to ambient. Also the  comes to mind. If there is such a prediction of the possibility of hydrogen remaining metallic at ambient, is there an analogous example of this with a material I'm more familliar with, or is this something quite new? Getting user provided data out of a database and into R I have a very simple question. When I do this:  \catcode`\@=0 \catcode`\\=11 @mycommand{a} @catcode`@\=0 \catcode`\@=11   It works perfectly and calls \mycommand correctly. However, when I try to define \mycommand with similar trickery:  \newcommand\mycommand[1]{   \catcode`\@=0   \catcode`\\=11   @textbf{Hello #1}   @catcode`@\=0   \catcode`\@=11 }   compilation stalls and gives me the * prompt. If I give it Ctrl+D, it terminates without producing any output. Indeed, it seems that changing the catcodes inside a command has no effect whatsoever, because the following works perfectly (whereas, from my understanding, it shouldn't):  \newcommand\mycommand[1]{   \catcode`\@=0   \catcode`\\=11   \textbf{Hello #1} }   Can anybody please explain to me what's happening? Why does 64-bit Windows need a separate "Program Files (x86)" folder? I'm using Mathematicato produce some of my graphics (when i'm not able to just TikZ them easily). All these images are availabvle as .eps and .pnghaving the same file name - let's use img.eps / img.png in the following.  I would like to set a preference for graphicx to use only one type, i.e. using  \includegraphics[width=.6\textwitdh]{img}   I would like to specify, which img to use with one switch, (package) option or variable.  I tried  \DeclareGraphicsExtension{.eps}   to restrict the allowed image extension to .eps, but still the .png file is included. It seems, .pngis preferred over .eps.  Though this is of course restricted to XeTeX it might be interesting for others, if you use .png together with some lossy format like .jpg to have a file extension preference.   Is there a package option for that or an easy command (global document option) for that?  To provide an MWE is - I think - difficult, because it would be necessary to provide at least 3 files. How do I delete all empty directories in a directory from the command line? Say I have a directory named foo/. This folder includes subdirectories. How can I delete all the empty directories in one command? I have a production server running with the following flag: -XX:+HeapDumpOnOutOfMemoryError  Last night it generated a java-38942.hprof file when our server encountered a heap error. It turns out that the developers of the system knew of the flag but no way to get any useful information from it.  Any ideas? What are the odd of a single coin toss after many consecutive ones? I have little knowledge of probability and have recently been thinking of a math problem I am sure you guys could answer  If for example I tossed a coin ten times and got all heads, would the odds of the next coin toss beeing heads be 50 50 or s very low as the odds of getting 11 in a row is tiny?   Thanks (Almost) double light speed Elementary central binomial coefficient estimates How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}&lt;\binom{2n}{n}&lt;\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ &gt; 1 ?     Does anyone know any better elementary estimates?   Attempt.  We have $$\frac1{2^n}\binom{2n}{n}=\prod_{k=0}^{n-1}\frac{2n-k}{2(n-k)}=\prod_{k=0}^{n-1}\left(1+\frac{k}{2(n-k)}\right).$$ Then we have $$\left(1+\frac{k}{2(n-k)}\right)&gt;\sqrt{1+\frac{k}{n-k}}=\frac{\sqrt{n}}{\sqrt{n-k}}.$$ So maybe, for the lower bound, we have $$\frac{n^{\frac{n}{2}}}{\sqrt{n!}}=\prod_{k=0}^{n-1}\frac{\sqrt{n}}{\sqrt{n-k}}&gt;\frac{2^n}{\sqrt{4n}}.$$ By Stirling, $n!\approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$, so the lhs becomes $$\frac{e^{\frac{n}{2}}}{(2\pi n)^{\frac14}},$$ but this isn't $&gt;\frac{2^n}{\sqrt{4n}}$. Possible Duplicate:          There are several tarballs (e.g. Firefox, Eclipse, Zend Studio and ...) that have executable files in them, that we can extract and run. but this executable will be available only for the user that extracts it.  If we want to make it available for all users we should move it to some location like /usr/share, /usr/local/share/, /opt/, ... and give appropriate permissions, make an executable file in /usr/bin/ and so on...  I want to make a deb installer file that does them all! How to do it? How do I complete the square from the point I have left off at, and is this correct so far?   I have a normal prior for $\beta$ of the form $p(\beta|\sigma^2)\sim \mathcal{N}(0,\sigma^2V)$, to get:  $p(\beta|\sigma^2)=(2\pi\sigma^2V)^\frac{p}{2}\exp[-\frac{1}{2\sigma^2}\beta^T\beta]$  where $\beta^T\beta$ is   $\sum\limits_{i=1}^p \beta_i^2$.  My likelihood has a normal distribtuion for the data points y of the form $p(y|\beta,\sigma^2)\sim\mathcal{N}(B\beta,\sigma^2I)  $  $p(y|\beta,\sigma^2)=(2\pi \sigma^2V)^\frac{n}{2}\exp[-\frac{1}{2\sigma^2}({\bf y}-{\bf B}{\bf \beta})^T({\bf y}-{\bf B}{\bf \beta})]$  (Note that $\beta$ is a matrix/vector too, \bf does not work.)  To get my posterior for $\beta$ I combined the above, took the exponential parts only, and then expanded to get:  $\exp[-\frac{1}{2\sigma^2}({\bf y}^T{\bf y}-{\bf y}^T{\bf B}\beta-\beta{\bf B}^T{\bf y}-\beta^T{\bf B}^T{\bf B}\beta)]\exp[-\frac{1}{2\sigma^2}({\bf \beta}^T{\bf B})]$.  I dropped the $({\bf y}^T{\bf y})$ term, as is not a function of $\beta$.  Putting into one expression without the exponential:  $-\frac{1}{2\sigma^2}(-{\bf y}^T{\bf B}\beta-\beta{\bf B}^T{\bf y}-\beta^T{\bf B}^T{\bf B}\beta+{\bf \beta}^T{\bf B})$.  I know I need to combine the similar terms and get into the form of the multivariate normal distribution, which is what I am aiming for, but I am unsure how to do this? I probably have to add an extra term to the expression to get it into the correct form?  Note: This is not homework, it's a project, but my Bayesian working knowledge is not good at all and so I need to understand the working out. I intend to integrate out the $\beta$ and then the $\sigma^2$ after getting into the multivariate form. Simple problem with Manipulate and Plot Questions about a commutative ring with exactly three ideals Let $R$ be a commutative ring with identity. Assume that $R$ has exactly three distinct ideals: $\{0\},I, R.$      1) Show that if $a \in R-I$, then $a$ is a unit in $R$.      2) Let $a,b\ne0$ in $I$. Show that $ab=0$. Check if user is using IE .NET OpenSource projects and strong named assemblies? Where did the world go? I was playing on my main world when my laptop ran out of power.  I found a charger and woke it up, but Minecraft had crashed.  Now, it doesn't appear in the world selection screen!  I have verified that it still exists in the Minecraft folder, and the directory contents look to be in order.  Does Minecraft save an internal flag somewhere when it crashes hard?  If so, how do I reset it? How do I prevent "Read more: www.site.com" from being injected with non-Tynt JavaScript into copy-paste actions? There are many sites that insert something like "Read more: www.site.com" into your clipboard when you copy text in your browser.  I find this abominable.  A google search reveals   to stop this if the website happens to be using the service  (by just blocking the Tynt domain), but many websites use home-grown JavaScript.  Is there a general way to block this behavior besides just turning off JavaScript?   I use Chrome but I am interested in all solutions.  EDIT: To be clear, I want to be able to keep the rest of the JavaScript functionality for these websites, since many will break without it.  EDIT #2: Here are two example websites which continue to harass me despite my use of a Tynt blocker:  |   Here are two   and two   which explain ways to implement this shady practice manually without Tynt.  Here is a  describing how hard this turns out to be to block reliably.  Here is the  I could find (March 2013) which offers suggestions on how to block this in Chrome using AdBlock, but it didn't work for me. How do I add a .sty file to my MacTeX/TeXShop installation? I've searched online, but I have not found any clear instructions how to permanently add .sty files to my TeXShop installation. I'm using a Mac with OSX (TeXShop is a mac only program).  What is the best way of doing this? The one comprehensive set of instructions I found .   I've got as far as extracting the files (running LaTeX on the .ins and . dtx files) but after that I don't know what to do. I also don't understand how to update the index of my distribution, and whether there are particular OSX specifics (the instructions are for Windows).