In his book, Reverse Mathematics, John Stillwell says (pg. 44) "that each continuous function on R may be encoded by a set of natural numbers and hence the arithmetization project extends at least as far as the continuous functions. This remarkable result is due to Borel (1898), p. 109, and it follows that each continuous function may be encoded by a real number."  The referenced book by Borel is in French.  I can follow that a continuous function on R may be encoded by a set of natural numbers. But how does one go from that set to a real number specifically?  Of course I tried to search for this result on the innertube but, alas, I got nowhere. I'm not even sure what category in mathematics to start with to narrow the search down.  Also, given the real number, is it possible to reverse the encoding to get the continuous function back? Does every real number represent a continuous function? What book(s) are available to learn more about this intriguing result?
I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it?
Short story in book collection about war machines and armies out of control I read in an anthology book in 1975. It starts with two people watching a “robot” plane glide high and then dive into an opening of a factory causing a massive explosion. I’m not sure but I think the people may have orchestrated the attack on purpose to get the robot armies fighting each other in the hope that it would diminish robot army capabilities and bring them under control. The robot wars did occur, but at the end of the story the people discover a tube sticking out of the ground that is periodically shooting something into the air. Nearby they find a canister that did not get launched out of the “cannon” successfully and it was broken open. After close examination they discover tiny “ants” on the canister that are actually tiny robots which are rebuilding the robot war factories. The robots have adapted and will keep trying to fulfill their mission. This story may  have been in a booked edited by Roger Elwood, but I cannot guarantee it. I added a comment to  but something strange is happening. The end code reads "); but SO decided to change that to ";);.    I did not add a ; in front of the ).  To further explain what's happening here, the following:  "http://example.com/." http://example.com/. "http://example.com." http://example.com. "http://example.com" http://example.com   renders as: How to insert Bengali text in IEEE conference paper? I am writing a paper for an IEEE conference using IEEEtran.cls. The primary language of the paper is English but my work deals with Bengali language. I want to know how do I insert some Bengali words such as পারে in the paper. Add/Delete table rows dynamically using JavaScript What is the connection between Poisson brackets and commutators? The Poisson bracket is defined as:  $$\{f,g\} ~:=~ \sum_{i=1}^{N} \left[  \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}} \right]. $$  The anticommutator is defined as:  $$ \{a,b\} ~:=~ ab + ba. $$  The commutator is defined as:  $$ [a,b] ~:=~ ab - ba. $$  What are the connections between all of them?  Edit: Does the Poisson bracket define some uncertainty principle as well? How to list all installed packages I'd like to output a list of all installed packages into a text file so that I can review it and bulk-install on another system.  How would I do this? Which attacks are possible against raw/textbook RSA? I would like to know where to find the source code of Ubuntu. I'd like to see how far it is "open source". "Do you want to continue[Y/n]?" Why the upper case? I am new to Ubuntu and some little thing start to bug me. Such as when trying to install a program, Ubuntu asks,     Do you want to continue[Y/n]?   What is bugging me is why the letter "Y" is upper case and "n" is lower case. Do they have any significant reason why one is upper and the latter is lower? Whatever case I enter it doesn't change the outcome: it either continues or not. Are they trying to notify us that we can enter both and it won't change anything? Commas with "so" I am confused with the usage of so.  Consider some examples from Oxford and Meriam-webster        It was still painful so I went to see a doctor (Oxford).   We were bored with the movie, so we left (MW).   There are no more chairs available, so you'll have to stand (MW).      So is a conjunction in all the above sentences. But I don't understand when to place a comma before it. So is one of FANBOYS. Thus, we have use a comma when it connects two independent clauses, which I think contradict the first sentence.  Is the placement of comma before so depends on a style guide? For example, Americans prefer a comma while the British doesn't. To reopen a closed question,  must agree that the question is suitable for the site and cast votes to reopen the question. But:  How does one actually vote to reopen? Is there a reputation level that you must have before you can see this function? Or do I just leave a comment along the lines of, &quot;I vote to reopen&quot;? What happens if someone with less than 3000 reputation leaves a comment that they wish to vote to reopen? How can one draw attention to their closed questions? Is it done via a posting new question? Or do you edit the closed question?   Related:  For more information, see &quot;&quot; in the . "viruses" or "virii"? Let $C$ be a finitely generated $R$-algebra where $R$ is unital commutative ring. Let $S$ be a simple $C$-module. Show that the annihilator of $S$ in $R$ is a maximal ideal. What I have tried is: When $C=R$ then it is a direct consequence of the simplicity of $S$ as an $R$-module. Further, since $S$ is simple then $S$ is cyclic (i.e. $S=Cy$ for nonzero $y\in S$). Consider the map $$f: C \to S$$ $$c \mapsto cy$$ Then $\ker(f)$ is the annihilator of $S$ in $C$ and $\ker(f) \cap R$ is the annihilator of $S$ in $R$. I have shown that it is an ideal of $R$ but I have no idea to continue showing the maximality. Is it a true direction or is there anything else I should consider to show the maximality? Determinant of a square matrix in a field \begin{array}{rrrrr|r}     b &amp; a &amp; a &amp; \cdot \cdot \cdot &amp; a \\     a &amp; b &amp; a &amp; \cdot \cdot \cdot &amp; a \\     a &amp; a &amp; b &amp; \cdot \cdot \cdot &amp; a \\     \cdot &amp; \cdot &amp; \cdot &amp; \space &amp; \cdot\\ \cdot &amp; \cdot &amp; \cdot &amp; \space &amp; \cdot\\ a &amp; a &amp; a &amp; \cdot \cdot  \cdot &amp; b   \end{array}  I have the above matrix $A\in M_{n\times n}(F)$ where $F$ is a field and $n\geq1$, $a,b\in F$.  I'm trying to find out how to use row operations to make it into an upper triangular matrix in order to figure out the determinant. But I'm not sure how I would approach it. Trying to dual boot Ubuntu 14.10 with Windows 10, last one not detected I have a laptop, and I'm going to install Ubuntu (I have experience, and I have  successfully installed , Ubuntu, , , , , , etc.), but installing  (Utopic Unicorn) alongside , Ubuntu can't detect Windows.  It shows me like there's no OS, but I have Windows&nbsp;10.  I have two primary partitions by the way if I manually install. If the Ubuntu installer can't detect Windows&nbsp;10, will it be detected at ? How do I install Ubuntu from an USB drive / create a live USB-stick in Windows? I tried to use the usb-creator.exe from Ubuntu 10.10 desktop i386 ISO image and it does create an USB stick that doesn't boot. When I try to boot from it I get:     Operating System not found.   I try to do this from Windows 7 x64. I'm not looking to create a live Ubuntu media, only an installation USB disk.  I repeated the process several times and I even tried to reformat the USB drive (FAT32), no success yet.  The USB stick is a 8GB one. For ridiculously small hard drives (8GB), what Ubuntu Based (although not necessarily official) distribution offers the smallest installed size?  Let me clarify: I still want a certain amount of personal file space, preferably a few gig. I also don't have a problem with performance, while it is mediocre, a 2.6GHz single-core and 1GB of ram suffice to run ubuntu fine, but i find that it  uses 6-7GB of space. I am aiming for closer to 4GB. I am working on a print for my wall and I am trying to create something like this:      I have already created the letters in Illustrator and now I am adding the colors in Illustrator and then exporting everything to Photoshop and then using the liquify tool to blend the colors to create a similar effect. I also used the healing patch tool to blend in the hard edges.  Is that the right way of doing it? Or is there another way? What is the list of keys and their corresponding wins?  I have it listing that the highest number of wins I've had in arena is a Champion key, but it doesn't say how many wins it was. Remove rogue app from menu bar