Can this font be implemented in LaTeX?
Old-style/Antique typesetting in LaTeX/TeX
Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits. $$\lim_{x\to 0}\frac{10^x - 2^x - 5 ^ x + 1 } {x\tan x} $$ This is an indeterminate limit.  I want help in solving this problem. Thanks in advance If $G$ is cyclic, then $G/H$ is cyclic?   The proof I got goes like this: $G$ is cyclic, so $G=&lt;g&gt;$ for some $g\in G$. So any coset in $G/H$ would be of the form $Hg'=Hg^n$ for some $n$. So $Hg$ is an generator of $G/H$. Thus, $G/H$ is cyclic.  I might just be confusing myself, but we have only shown that $Hg'$ is in form of $(Hg)^n$. But what if we are missing some $n\in\mathbb{N}$? That is, there is no quotient group of the form $Hg^2$, for example.  To make myself a little bit clearer, I think what the above proof has done was showing that $\forall Hg'\in G/H, Hg'\in &lt;Hg&gt;$, thus $G/H \subset &lt;Hg&gt;$. I feel that this is not a complete proof. I've seen this Kuratowski definition for ordered pairs, but can't fathom why it implies an order to $x$ and $y$  $(x,y):=\{\{x\}, \{x,y\}\}$  As I understand sets, $\{\{x\}, \{x,y\}\}$ is also $\{\{x,y\}, \{x\}\}$. Only when I think about the Axiom of Union does $\{\{x\}, \{x,y\}\}$ "collapse down" to $S = \{x, y\}$, but that doesn't give me much either. All I can see is some as yet hidden message in the set saying "I am the set $\{x,y\}$ and my order of $x$ first is indicated by having $\{x\}$ along for the ride." Is the Dungeon Magazine 4e Adventure List spreadsheet still available? used to go to a spreadsheet with all the 4E adventures published in Dungeon Magazine (and possibly some others, IIRC) during that edition's duration. I remember it listed out each adventure by name, start/end level, adventure path, and maybe even a little summary blurb.  However, the spreadsheet behind that link seems to have disappeared into the digital ether, and I can't find a copy of it in my own google drive (implying I never made a copy of it).  Does anyone happen to have a copy of that spreadsheet that they could share, and/or other resources that would let me sort through the many adventures from 4E Dungeon Magazine? I have installed the pimd service by means of apt. This comes with an upstream systemd unit file (/lib/systemd/system/pimd.service).  I want the service to be restarted when for some reason it gets killed, hence I wish to add the line Restart = always in the unit file.   However, I don’t want to modify the upstream unit file.  Is there any workaround for this? Why are there no line numbers in the code listings? Would it make discussing code easier if you could simply refer to a particular line or range, or am I wrong? Can EU citizens travel to UK after Brexit with just an identity card? Plotting several 2d functions in a 3d graph I am trying to reproduce a similar figure to the following one. My first guess was to use pgfplots but I didn't find any example that I could use as a starting point. At the moment I am unsure if pgfplots is actually the right package or I should rely on something else e.g. TikZ.   Since I don't really know where to start, I would appreciate any piece of code that points me in the right direction.    EDIT  Here is some dummy data for the plots.  plot1   plot2   plot3   plot4 0       0       0       0 3.466   2.058   0       0 4.262   2.976   0.001   0 3.822   3.168   0.006   0.008 2.953   2.936   0.019   0.063 2.065   2.492   0.046   0.265 1.332   1.977   0.092   0.734 0.797   1.478   0.164   1.508 0.443   1.045   0.268   2.44 0.228   0.698   0.412   3.219 0.107   0.438   0.598   3.524 0.046   0.256   0.831   3.219 0.017   0.138   1.109   2.44 0.006   0.067   1.429   1.508 0.002   0.029   1.78    0.734 0       0.01    2.141   0.265 0       0.003   2.479   0.063 0       0.001   2.736   0.008 0       0       2.808   0 0       0       2.465   0 0       0       0       0 Inspired by :  In my answer, I had stated that Patronuses appear to not be physical beings because they have no effect on anything other than Dementors.     They don't cause leaves to stir in the movies. They passed straight through the tent. They don't appear to be bound by normal laws pf physics and can travel great distances almost immediately (Shacklebolt was presumably in London when his Patronus issued the warning). They can apparently walk across water without sinking (as they did in The Prisoner of Azkaban). They can jump, walk, and float in mid-air (as seen in the Room of Requirement in the movie version of The Order of the Phoenix). They make no footprints in lightly powdered snow (The Deathly Hallows Chapter 19) and make no sound of crunching snow as Harry does (further down the same page in Chapter 19).   I stated incorrectly that the only place in canon where there is an exception is with Dementors, where the Patronus physically fends them off with its horns.  A Dementor is clearly a physical being, if it prised Dudley's mouth open and (in the movie anyway) physically slid open the train door.)   Based on comments and further thought, I can think of at least two other examples when a Patronus interacts physically with a physical being.   As pointed out by Slytherincess, a Patronus can fend off a Lethifold (Living Shroud). When training, the Patonus took form against a Boggart, which had taken the form of a Dementor.   All three of these things have, AFAIK, only three things in common (but I may be off).   They are evil They are magical They are, indeed, physical beings.   So the question is, why can a Patronus interact with all three of these, but seems to have no physical effect on everyday things? Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$ I am looking for the combinatorial argument for the identity:  \begin{equation} k\binom{n}{k} = n\binom{n-1}{k-1} \end{equation}  This is easy to show algebraically as:  \begin{equation} \binom{n}{k} = \dfrac{n(n-1)(n-2)(n-k+1)}{k(k-1)!} \end{equation}   What is the combinatorial argument? What are some general ideas to get started?   Here is a clarification of 2. From what I have seen so far, proving (combinatorially) an identity with an addition sign usually implies that we need to partition a set (this makes sense because of the addition rule and provides a nice visual). On the contrary, the previous observation leads me to believe that multiplication in identities can be resolved with the multiplication principle, but what is the "visual/interpretation" for this? Could someone provide such an interpretation for the example identity given above? Likelihood Ratio Test for Exponential Distribution with a Limited Parameter Space Proving the measure of an increasing sequence of measurable sets is the limit of the measures Is there any way to see the custom messages/remarks of resolved/expired bounties? Force drop db while others may be connected Terminal not opening up after upgrading python to 3.7 I have upgraded python to version 3.7 and now my terminal is not opening up.  I followed this post   Commands I ran are as follows,  sudo apt-get install python3.7 sudo update-alternatives --install /usr/bin/python3 python3 /usr/bin/python3.6 1 sudo update-alternatives --install /usr/bin/python3 python3 /usr/bin/python3.7 2 sudo update-alternatives --config python3   Entered 2 for python3.7  I can't open terminal and run commands.  My ubuntu version is, Ubuntu 18.04.2 LTS I was wanting to initialize some strings at the top of my script with variables that have no yet been set, such as:  str1='I went to ${PLACE} and saw ${EVENT}' str2='If you do ${ACTION} you will ${RESULT}'   and then later on PLACE, EVENT, ACTION, and RESULT will be set. I want to then be able to print my strings out with the variables expanded. Is my only option eval? This seems to work:  eval "echo ${str1}"   is this standard? is there a better way to do this? It would be nice to not run eval considering the variables could be anything. Can PostGIS and QGIS provide dependable multi-user editing? After considering a few options, I'm trying to give my 15-20 QGIS 2.2 users the ability to concurrently view/edit one or two basic point PostGIS layers, largest one has just 16,000 records. I've installed PostgreSQL 9.1 / PostGIS 2.0 and basic testing seems to work ok but haven't unleashed it to the masses yet.  We'll typically only have 2-3 editors at a time, but possibly 15 viewers. Edits will all be in QGIS, viewers will be QGIS and ArcGIS (but we don't have SDE).  As I said, these are just basic point files, and  edits will just be adding/deleting points and some table edits. I am not worried about editing the same features at the same time.  Can the combination provide dependable multi-user editing for two basic point files? Story featuring smart people sent on a space mission to nowhere The story in question featured, as the vague title suggests, a "space mission" that is really a cover for forcing a bunch of really smart people to be in close proximity to each other for long enough that they can truly evolve their collective intelligence. I recall them developing new maths and, e..g, "farseeing" and eventually figuring out that while there was no planet at the end of their journey as they were told, they were able to construct one for themselves. I have designed a circuit where I want to give a pulse of very short duration using a switch. How can I implement it in spice.