Diagonal ladder length Let  $ ABCD $ be a square with side length $ a $. Let $ s $ be a staircase from $ A $ to $ C $ with total length $ l $ and number of steps $ n $. It consists of perpendicularly alternating lines of length $ \frac{a}{n} $, as .  We see that $ l $ can be expressed as follows: $$ l = \frac{a}{n} \cdot n + \frac{a}{n} \cdot n = \frac{a}{n} \cdot n \cdot 2 = 2a $$ and as such stays constant at $ 2a $.  Now let us imagine that the amount of steps is infinite, e.g. $ n =  \infty  $. Per definitionem, the staircase should now be the diagonal of the square with length $ l=a\sqrt{2} $ according to Pythagoras. This is paradoxical! According to the equation pictured above, it should have the length $ 2a $, not $ a\sqrt{2} $.  My question is:  Does $ l $ equal to $ 2a $ or $ a\sqrt{2} $?
The staircase paradox, or why $\pi\ne4$ What is wrong with this proof?    Is $\pi=4?$
C# - Keyword usage virtual+override vs. new What are differences between declaring a method in a base type "virtual" and then overriding it in a child type using the "override" keyword as opposed to simply using the "new" keyword when declaring the matching method in the child type? Why copy constructor not invoked here Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ Deriving the Coulomb force equation from the idea of photon exchange? Since Newton's law of gravitation can be gotten out of Einstein's field equatons as an approximation, I was wondering whether the same applies for the electromagnetic force being the exchange of photons. Is there an equation governing the force from the exchange of photons? Are there any links which would show how the Coulomb force comes out of the equations for photon exchange? I know that my question is somewhat similar to the one posted here , but it doesn't really have an answer to my question specifically. Why won't Mathematica display an answer? Is the set of all unitary matrices in $M_2(\mathbb{C})$ is compact? I can show that as determinant map is continuous so unitary matrices are closed but how to show they are bounded?   Please help. Video editing software options? What video editing software would you recommend on Ubuntu (or Linux in general) and why?  It can be for either beginners or experienced users.  Please only place one piece of software per response and include details to what makes it great! I am often logged in several SSH sessions at once. To logout from several sessions, I press CTRL+d, until I am back on my local machine. However, I occasionally press it once too many, and my terminal exits. Is there a way to make CTRL+d unable to close my terminal ? I am using terminator as my terminal emulator. I'm looking for a good algorithms for the following problem: Given a 3D grid of voxels (which may be either empty or filled), if I pick two non-adjacent voxels, I want to know if they are connected to each other by other voxels.  For example (to illustrate the situation in 2D), where # is a filled square:    1 2 3 a # # # b # # c # # #   If I pick a3 and c3, I want to determine as quickly as possible if they they are connected; if there is a path between a3 and c3 through filled pixels. (The real situation is in a 3D voxel grid, of course.)  I've looked at floodfill algorithms and path finding algorithms, but I'm not sure which to choose. Both do unneccesary work: Flood fill tries to fill all voxels, but this is not needed. Path-finding algorithms are usually concerned with finding the shortest path, which is also not necessary. I only need to know if there is a path.  What algorithm should I be using?  Edit: based on the comments, I think should add the following: the content of the voxels are not known in advance, and also, the algorithm is needed to detect if removal (emptying) of a voxel would cause the group of voxels to break into two or more smaller groups. What to do with an additional answer which IMHO explains only a small portion of the question, while another answer explains more For the apple.stackexchange question  a 2nd answer was given, which relates to the question, but IMHO does not answer a majority portion of the question. Here some details:   The question is about, how to set the PATH variable in a unified manner. The  details how to change the file /etc/launch.conf. It is to note, that changing this file is needed for the overall answer. The author of the answer does make the valid point that, when he searched for, how to change /etc/launch.conf, the question came up. The  does not detail the change of /etc/launch.conf enough.   So, the general problem is: What to do when an additional answer explains only a small (but unique portion) of the question, while an earlier answer explains more (but not what the additional question explains).  Finally, I think I should note that I am the initial asker and poster of the 1st answer. Let $\displaystyle \sum a_n$ be  convergent series of positive terms and set  $\displaystyle b_n=\sum_{k=n}^{\infty}a_k$ , then prove that $\displaystyle\sum \frac{a_n}{b_n}$ diverges.   I could see that $\{b_n\}$ is monotonically decreasing sequence converging to $0$ and I can write $\displaystyle\sum \frac{a_n}{b_n}=\sum\frac{b_n-b_{n+1}}{b_n}$, how shall I proceed further? I have quite a bad habit of copy-and-pasting my packages from one LaTeX document to another. Is there any way we can detect unused packages (i.e. \usepackage commands) so that they can be removed from the LaTeX document? Bug Identification: Tiny Bugs How do I design a 2A or more power supply for my consumer USB devices? I understand that many of my consumer devices charged by USB ports can charge at higher rates than 0.5A.  However, in testing, I've found they don't consume more than this. Test Setup  I have a power supply that outputs between 0 and 30 volts, and 0 to 20 amps. I set the voltage to 5 volts, then connected up a female USB to the wires correctly. When I plug in a Samsung Galaxy S4 Mini, it pulls around 0.44 amps. It is the same with a Samsung Galaxy Tab Pro. However, when I connect an iPad Air with an official lightning cable, it only pulls around 0.11 amps. Also, I tried connecting a ZGPAX S28 smartwatch and it was still about 0.44-0.45 amps.  The power supply is more than capable of outputting the maximum 2.1 amps, so why doesn't it to at least the tablets? What do I need to do in my test setup to convince the devices to consume their maximum charging current? Does Wasserstein distance require the source and target distributions to have the same mass? If we minimize the Wasserstein loss, $$W_1 (P_S, P_T) = \underset{\gamma \in \Pi}{\text{min }} \sum_{x,y} |x-y|\gamma(x,y)$$ which means we are looking for the coupling $\gamma(x,y)$ that minimizes the cost of moving mass from the source distribution $P_S$ to the target distribution $P_T$,  Do the masses of the source and target distributions have to be identical? If so or not, how are the masses of $P_T$ and $P_S$ accounted for mathematically in, or when minimizing, the shown formula, or what might be missing/overlooked by the Wasserstein approach overall due to the mass differences? My question is, if you create a new domain user account for each of the SQL Server processes, what permissions should be set for each account? Or does the SQL configuration manager actually take care of this, and I just had an unforeseen issue?  I quite often have to setup Microsoft SQL Server and wondered if anyone can provide advice on configuring the accounts the services should run as. IMO this has been vaguely documented by Microsoft, while they point you in the right direction I have never been able to find any concrete examples.  To summarize what I've seen so far:  For simple deployments\development environments it is OK to use the virtual account defaults the installer uses: e.g. NT SERVICE\MSSQLSERVER  Avoid using the SYSTEM account, this is not secure.  For production and in domain environments it's recommended to use either a Managed Service Account, or create a domain user account (not an admin) for each service. Allegedly if you use a domain account at installation time the installer will set any required permissions for you.  If changing the service account on an existing install from a virtual account to a domain account the recommendation is to use the SQL Server configuration manager to set the new service accounts. Allegedly this will set any required permissions for you.  I just tried changing the service account in an existing install to a domain account and it would give me a logon failure until I granted the account log on as service permission, which contradicts the part where the SQL Server configuration manager will set any required permissions. (Although I'm not sure if a GPO may have interfered with setting this local security policy)  Microsoft does provide a list of the permissions that the SQL Server Setup grants .  But it's not clear to me if that is something I should be doing manually for the user I create to run the service as, or whether using the SQL config manager should automatically set these permissions.  SQL Server 2014, Domain Controller is on Windows Server 2008 R2. Scale and Rotate Stopped Working [2.8] Wifi hotspot software package other than ap-hotspot? I have tried using ap-hotspot to create a wifi-hotspot but it gives me some problem every time I use it . I just want to ask is there any other software  package that create a wifi-hotspot othen than ap-hotspot ? How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$? Multiple bibliographies (reference lists) using embedded bibliography database I would like to have multiple bibliographies (reference lists) as described in   ().   However, I'd like to use with a single .tex file, which would contain the bibliography database embedded, as in  ...   Now, I want to use letter documentclass, for which the bibliography needs to be defined ()   So what I end up is with this example (code below, compiled with just pdflatex, no bibtex(which I'd like to avoid in this case)):     ... obviously, something works (multibib recognized primary and secondary cites, assigning them number "1" as start) -- however I don't get two, separate primary and secondary list of references; would that be possible in this context? (doesn't have to be with multibib, but I'd like to avoid a separate .bib file and a bibtex call)  EDIT: I'd also like separate cite keys (i.e. rendered as [a1] and [b1]; not yet sure if multibib can do that; edit: no it cant,  says "You can also edit the .bbl files created by BibTEX and add 'a', 'b',. . . to the labels, but your changes will be overwritten by subsequent BibTEX runs." and recommends \cite[a]{key}) - how possible would that be?  &nbsp;  The code is this:   \documentclass{letter}  % Environment thebibliography undefined when using letter: http://www.latex-community.org/forum/viewtopic.php?f=4&amp;t=3359 \makeatletter \setlength \labelsep {.5em} \newcommand\newblock{\hskip   .11em\@plus.33em\@minus.07em} \let\@openbib@code\@empty \newcommand\refname{References} \newcommand\section{\@startsection   {section}{1}{\z@}%   {-3.5ex \@plus -1ex \@minus -.2ex}%   {2.3ex \@plus.2ex}%   {\normalfont\Large\bfseries}} \newenvironment{thebibliography}[1] {\section*{\refname}%   \@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}%   \list{\@biblabel{\@arabic\c@enumiv}}%   {\settowidth\labelwidth{\@biblabel{#1}}%     \leftmargin\labelwidth \advance\leftmargin\labelsep \@openbib@code     \usecounter{enumiv}%     \let\p@enumiv\@empty \renewcommand\theenumiv{\@arabic\c@enumiv}}%   \sloppy \clubpenalty4000 \@clubpenalty \clubpenalty   \widowpenalty4000%   \sfcode`\.\@m} {\def\@noitemerr {\@latex@warning{Empty       `thebibliography' environment}}%       \endlist} \makeatother  \usepackage{multibib} \newcites{pri}{Primary sources} \newcites{sec}{Secondary sources}  \begin{document}   \begin{letter}{Addressee} Blah, blah - content...  Primary cite: \citepri{lamport94} ...  Secondary cite: \citesec{lamport94} ...   \begin{thebibliography}{9}  \bibitem{lamport94}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1994.  \bibitem{lamport95}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1995.  \end{thebibliography}  \end{letter}   \end{document}