This is an extension question of .  They have given a plenty nice way to fold a A4 sheet in 3 equal parts.   What is the Mathematics behind the foldings? Can we use the same way for $A5,A6,\ldots$?
This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on eye estimate where to fold. Then I turn the letter over and fold bottom to top. Most of the time ending up with three different areas. There must be a way to do this exactly, without using any tools (ruler, etc.).
How can I send an email by Java application using GMail, Yahoo, or Hotmail? Is it possible to send an email from my Java application using a GMail account?  I have configured my company mail server with Java app to send email, but that's not going to cut it when I distribute the application. Answers with any of using Hotmail, Yahoo or GMail are acceptable. Concurrent updates to GDAL (to 2.1.3) and QGIS (to 2.18.5) on Linux Mint 18.1 from the Ubuntugis repos breaks QGIS, with a noted dependency for the virtual package gdal-abi-2-1-2 being uninstallable.    The stable Ubuntugis ppa currently supplies GDAL 2.1.0 (gdal-abi-2-1-0) while the Ubuntugis-unstable ppa has been updated to GDAL 2.1.3 (gdal-abi-2-1-3).  Has anyone found a way around this?    Update:  QGIS bug report for those interested is here:     UPDATE 2  The ticket has been closed as fixed. Everything should be back to normal now. theorem numbering: section counter but remove the section number Is there an open dense set $S \subset [0,1]$ such that $m(S)<1$? $m$ is the Lebesgue measure. I was thinking that: $\Bbb Q$ is dense and $m(\Bbb Q)=0&lt;1$ but it fails to be open, but maybe I could construct an open set from this fact, or also using that $(0,1)$ is a dense and open subset but $m(0,1)=1$.   Any hints or ideas will be very appreciated. Thanks. I find in many places mentions of pre-amplifiers being used to boost audio signals prior to feeding to a power amplifier.  Wikipedia's article on  gives some hints at their usage, but it doesn't really explain why it would be necessary to have two amplifiers in series rather than a single one with greater gain.  If a pre-amp is just a power amp with a smaller gain, what's the point, and why the distinction between the two kinds? Is the distinction in function, or simply purpose? .NET String.Format() to add commas in thousands place for a number Derive equation for a cantilever in SHM Do superheroes/villains like Quicksilver or Flash with super speed get tired due to running? Can toddlers be taught to safely navigate stairs? How to control the coordinates in Tikz & Matrix? I use Tikz with Matrix because I need to draw arrows and write on and beside the matrix. However, I don´t control the coordinates well other than using trail and error. For instance in the matrix below I have put an X at (0,0) and it is not in the centre of the matrix. However, if the matrix is 4x4 than it is in the centre. Is there any convenient way to know the coordinates? \documentclass[border=1cm]{standalone}  \usepackage{tikz} \usetikzlibrary{matrix}   \begin{document}  \begin{tikzpicture} \matrix(m) [matrix of math nodes, nodes in empty cells, nodes={minimum size=1cm, outer sep=0pt, text height=1.5ex, text depth=.25ex}] {  36     &amp;   &amp; 32    &amp;   &amp; 33    &amp;   \\         &amp;   &amp;       &amp;   &amp;       &amp;   \\  10     &amp;   &amp; 8     &amp;   &amp; 7     &amp;   \\         &amp;   &amp;       &amp;   &amp;       &amp;   \\  12     &amp;   &amp; 17    &amp;   &amp; 16    &amp;   \\         &amp;   &amp;       &amp;   &amp;       &amp;   \\  23     &amp;   &amp; 15    &amp;   &amp; 16    &amp;   \\     &amp;   &amp;   &amp;       &amp;   &amp;       &amp;   \\ };  \draw (m-1-1.north west) rectangle (m-8-6.south east);    % Lineas horizontales \foreach \i in {2,4,6,8} {   \draw (m-\i-1.south west) -- (m-\i-6.south east); }  % Lineas verticales \foreach \j in {2,4,6} {   \draw (m-1-\j.north east) -- (m-8-\j.south east); }  % dual ui \node [font=\color{red}] at (3.5,3) {$\ge 10$}; \node [font=\color{red}] at (3.5,1) {$\ge 8$}; \node [font=\color{red}] at (3.5,-1) {$=4$}; \node [font=\color{red}] at (3.5,-3) {$=3$};  % dual vj \node [font=\color{red}] at (-2.5,-5) {$\le12$}; \node [font=\color{red}] at (-0.5,-5) {$=12$}; \node [font=\color{red}] at (1.5,-5) {$=6$};  \node [font=\color{red}] at (0,0) {X};  \end{tikzpicture}     \end{document} I installed the latest Ubuntu today, 15.04. Spotify has stopped working. How can it be made to work again?  The Unity bar flashes green 4 or 5 times and then stops flashing. There are no error messages. It was installed successfully and played well on 14.10 till the upgrade a few hours ago. It just does not appear on the screen. What should I do when Ubuntu freezes? All operating systems freeze sometimes, and Ubuntu is no exception. What should I do to regain control when...   just one program stops responding? nothing at all responds to mouse clicks or key presses? the mouse stops moving entirely? I have an    In what order should I try various solutions before deciding to pull the power plug?  What should I do when starting up Ubuntu fails?  Is there a diagnostic procedure I can follow? Confused about the possibility of different differentiable structures. In Loring Tu's "An Introduction to Manifolds" an atlas on a manifold is a collection of coordinate charts that are pairwise compatible and cover the manifold. A smooth manifold is defined to be a topological manifold together with a maximal atlas or differentiable structure. The maximal atlas is constructed by taking an atlas and appending all coordinate charts that are compatible with the atlas and using this he shows that any atlas on a locally Euclidean space is contained in a unique maximal atlas.  He also later shows that for any chart on a manifold, the coordinate map is a diffeomorphism onto it's image.   My question arises when Tu mentions in an aside that every compact topological manifold in dimension four or higher has a finite number of differentiable structures. What I don't understand is how it is possible for a smooth manifold to have more than one differentiable structure. Doesn't the fact that for any chart the coordinate map being a diffeomorphism coupled with the fact that composition of smooth functions is smooth mean that any two coordinate charts on a manifold are compatible and thus they must all belong to one maximal atlas? How is it possible to have another? Am I not understanding the word "maximal" or atlas or some other concept? Why does subsurf modifier cause weird deformation at the end of a cylinder? My problem is simple. I think that the strange deformations on the top are there because there are so many vertices in one face. How do I fix this while maintaining nice topology?  Although I triangulated the top polygon, the problem still persists. How can I change the field names (not the alias) in ArcGIS 10.2? I already found this page on ArcGIS support: &quot;&quot; but there is is not &quot;Defense Mapping&quot; on Customized Mode Categories In my ArcCatalog! Without access to Defense Mapping, how can I change the field names? How to implement a progress bar using the MVVM pattern Altium - create a mirror image of a board I want to create an identical motor drive board in the right and left sides.  I just finished layout for the right side and I am wondering what is the easiest way to create the mirror image of this board for the left side?  For example: the connectors on the "top" side would now be on the bottom of the other board.  All components and tracks will also switch sides, layer stack will be inverted, etc.  Is there any command sequence in Altium that will do this without me having to manually re-assign everything to the other layer? Gauss's Law with Moving Charges Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$ This is a homework question whereby I am supposed to evaluate:  $$\sum_{n=1}^\infty \frac{1}{n^2 +1}$$  Wolfram Alpha outputs the answer as   $$\frac{1}{2}(\pi \coth(\pi) - 1)$$  But I have no idea how to get there. Tried partial fractions (by splitting into imaginary components), tried comparing with the Basel problem (turns out there's little similarities), nothing worked. Name of the book/series based on magicians who tap into computer grid?