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Is There a Natural Way to Extend Repeated Exponentiation Beyond Integers? This question has been in my mind since high school.  We can get multiplication of natural numbers by repeated addition; equivalently, if we define $f$ recursively by $f(1)=m$ and $f(n+1)=f(n)+m$, then $f(n) = m \times n$.  Likewise, we get exponentiation by repeated multiplication. If $g(1)=m$ and $g(n+1)=mg(n)$, then $g(n) = m^n$.  In my high school mind it was natural to imagine a new function defined by repeated exponentiation: $h(1)=m$ and $h(n+1)=m^{h(n)}$.  These definitions only make sense for $n$ a natural number, but of course there are standard very mathematically satisfying ways to define multiplication and exponentiation by any real number. My question is this:  Can the function $h$ defined above also be extended in a natural way to $\mathbb{R}^{&gt;0}$?  The question is in the spirit of seeking an extension of $f(n)=n!$ to $\mathbb{R}$ and arriving at $\Gamma(x)$.  Let me focus the question, and attempt to make precise what I mean by "in a natural way." Take $h(1)=2$ and $h(n+1)=2^{h(n)}$. $h$ is now defined on $\mathbb{N}$, and $h(2)=4$, $h(3)=16$, $h(4)=2^{16}=65,536$ etc. Is it possible to extend the domain of definition of $h$ to all positive reals in such a way that  a) The functional equation $h(x+1)=2^{h(x)}$ continues to be satisfied for all $x$ in the domain.  b) $h$ is $C^\infty$. (Analytic would be even better but this seems maybe too much to hope for?)  c) All $h$'s derivatives are monotone.  These requirements are my attempt to codify what would count as "natural." I am open to suggestions about what would be a better list of requirements.  If such a function exists, I would like to know how to construct it; if it doesn't, I would like to know why (i.e. outline of proof), and if relaxing some of the requirements (e.g. just the first derivative monotone) would make it possible.  (If the function exists, I am also interested in the questions, "is it unique?" "Could we add some natural requirements to make it unique?" But my main query is about existence.) What is the proper way to explain the twin paradox? Prove the statement 'World is not flat' In my graph theory book exercise, I found a problem that:     Prove that the World is not flat using Mathematics     This picture is used in the exercise, but no idea of applying it.  I would have used the angles don't add to $180^\circ$, but the author is using a fourth city. What is the command to add newline linebreak in \sum \prod? I am sure this question or similar has been asked but I cannot find it by searching so adding it to make it more accessible: what is the command/environment to add newline linebreak in \sum \prod so the expression is on two rows instead of one line hard to read?    MWE  \documentclass{article} \usepackage{amsmath,amsfonts}  \begin{document}  $$\sum^{N_{cut}}_{i=1,\newline e_1\in C_i,e_2\not\in C_i} \prod_{j\in C_i} p_j-\sum^{N_{cut}}_{i=1,e_2\in C_i,e_1\not\in C_i} \prod_{j\in C_i} p_j\geq 0$$  \end{document} How can I get an underbrace and an overbrace to partially overlap in an equation? Sometimes I want to describe portions of an equation using \underbrace and \overbrace in such a way that the portions are not completely distinct, for example:          /-------\ x = A + B + C + D + E     \-------/   \---/      What is the best way to do this? How to peel hard boiled eggs easily? This is one of the most annoying duties in the kitchen, for me.  What are yours tips and tricks to peel hard boiled eggs easily? Varying behavior for possible loss of precision I am revising for an exam and have just realised that the euclidean algorithm questions in past exams are much harder than in homeworks! So i need some help please.  I have a question here, i already have the solution to it but i dont understand it that well, so i would like someone to please explain what is going on ect, and dont worry about avoiding giving me the full solution as i already have it so a full solution would be good!  Here is the question:  Use euclids Algorithm to find $gcd(172,20)$.  Hence solve $172a + 20b = 1000$ giving all solutions in terms of parameter t. List any solutions for which $a,b&gt;0$.  So it wants the answers in terms of parameter t, which is something i am not familiar with and cannot find in my books.   I have the solution in the solutions paper as   $$172= 8\times20+12 $$ $$20=1\times 12 + 8$$ $$12=1 \times 8 +4$$ $$8=2\times 4+0$$  thus $gcd(172,20)=4$  then   $$4= 12-8=12-(20-12)=2\times 12 - 20$$  $$=2\times(172-8\times20)-20$$  $$=2 \times 172 -17 \times20$$   multiplying by 250 we get   $$1000=500 \times 172 -4250 \times 20$$  so one solution is $a=500, b=-4250$  So, I understand everything up until here!!  Now it says,  Generally, $$a=500-t \frac{20}{4}=500-t$$            $$b=-4250+t \frac{172}{4}=-4250+43t$$  To have $a,b&gt;0$, we need $$500-5t&gt;0, (100&gt;t)$$ to have $a&gt;b$ we need $$-4250+43t&gt;0,(t&gt;98.8)$$$  Need $t=99$, $a=5$, $b=7$  Could someone please help explain what is going on in this last bit of working out? I am pretty good at euclidean algorithm usually but havent seen this type before an am panicking  bit as so close to exam! Any help appreciated how to find t and the other positive solutions would be great!  Please could you show me exactly how it is done here as my lecturer is quite strict with us using his exact methods. Many thanks Interview question: three arrays and O(N*N) Variance of a linear combination of vectors Areas versus volumes of revolution: why does the area require approximation by a cone? Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$.  Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder.  However, if we want to find the surface area, then we approximate it as part of a cone and the formula is $$\int_a^b 2\pi f(x)\sqrt{1+f'(x)^2} dx.$$  But if approximated it by a circle with thickness $dx$ we would get $$\int_a^b 2\pi f(x) dx.$$  So my question is how come for volume we can make the cruder approximation of a disk but for surface area we can't. How do I say what is the current UTC in eastern zone Calculating Statistics Based on Expression in QGIS? Is it possible to calculate the sum of polygon areas based on whether or not it has a value in another column?   For example:  In column 2 I have the area of polygons. I want to calculate the area of all polygons that contain 'AG' in column 1 or all polygons that are IN column 1. But calculate them separately so I can get the total area of each type in column 1. 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. How can Cooper communicate with his daughter before the time he went to space? In the movie Interstellar it's shown that after Cooper fell into the Black Hole, he communicates with his daughter Murph through some codes as Ghost. My doubt is how can he communicate to his daughter before he goes into space? I didn’t get the logic of that part. Can anyone please make it clear? At the moment, in major part because they are two separate post objects in the system, the excerpt and the body of a tag wiki count as two separate posts with regards to suggested edits in the queue. Mechanically, this is sane, but when it comes to handling the actual queue, it can be confusing. A body edit might consist of little more than a bed of links, which looks bad. But then an excerpt is shown that includes the rest of the data, and since excerpts can be included on the actual wiki page, it makes a lot more sense when they're taken in the same space.  And this is how they are composed, too. The actual edit interface for working on tag wikis has both the excerpt and the body on the same page. In a sense, the excerpt behaves a lot like the "title" of the wiki, if we make a parallel to normal post edits.  Thus, my suggestion (which is to say a suggestion that I repossessed from Robert Cartaino) is that when suggested edits are reviewed for tag wikis, that they be presented in a single unit, treating the excerpt the same as how a title is treated in a question edit. This way, relevant information that is crossed between the two isn't lost because they are reviewed separately. Proving or disproving inequality $ \frac{xy}{z} + \frac{yz}{x} + \frac{zx}{y} \ge x + y + z $ Intuition for Rayleigh PDF We have ground-truth data $\mathbf{x}^* = (0,0)^T \in \mathbb{R}^{2}$. Furthermore, we have $N$ measurements $\mathbf{x}_i \in \mathbb{R}^{2}, i\in \{1,\ldots,N\}$. We calculate the $N$ error vectors $\mathbf{e}_i = \mathbf{x}_i - \mathbf{x}^*$. We assume that the two components of $\mathbf{e}_i = (e_{x,i}, e_{y,i})^T $ are independent of each other and normally distributed with equal mean $\mathbf{\mu} = (0,0)^T$ and variance $\sigma^2$. The absolute errors $e_i = \sqrt {e_{x,i}^2 + {e}_{y,i}^2} $ follow a Rayleigh distribution, is that correct?  The probability density function (PDF) of the Rayleigh function is zero at a value of zero. Why is this the case (intuitively)? I would assume that our errors should most likely be 0 (i.e. PDF should be highest at 0)... 'apt-get' always fails with ERROR 404: Not Found over a package (flashplugin-downloader) the following error is preventing me from installing anything on Xubuntu 11.10. How can I fix it??  $ sudo apt-get install meld Reading package lists... Done Building dependency tree        Reading state information... Done meld is already the newest version. 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. 1 not fully installed or removed. After this operation, 0 B of additional disk space will be used. Do you want to continue [Y/n]? Y Setting up flashplugin-downloader (11.0.1.152ubuntu1) ... Downloading... --2012-02-06 21:23:36--  http://archive.canonical.com/pool/partner/a/adobe-flashplugin/adobe-flashplugin_11.0.1.152.orig.tar.gz Resolving archive.canonical.com... 91.189.88.33 Connecting to archive.canonical.com|91.189.88.33|:80... connected. HTTP request sent, awaiting response... 404 Not Found 2012-02-06 21:23:36 ERROR 404: Not Found.  download failed The Flash plugin is NOT installed. dpkg: error processing flashplugin-downloader (--configure):  subprocess installed post-installation script returned error exit status 1 Errors were encountered while processing:  flashplugin-downloader E: Sub-process /usr/bin/dpkg returned an error code (1) Who made the decision for Gandalf's promotion from 'grey' to 'white'? In the first book of the LoTR, after his fight with the Balrog of Moria, Gandalf the Grey changed to .   My question is, who made the decision to 'promote' him from grey to white? Was it ? Or was it the ? Or was it the ?