Why doesn't TeX have a debugger? And by 'debugger' I mean a dedicated application in which you can set breakpoints and watch the values of registers, macros, etc. as they change in real time, maybe even preview, say, what a box register would look like if it was rendered right now, and try out what an ad-hoc TeX expression would expand to right now, and step through a macro expansion one expansion at a time?
..and I don't mean finding out why it doesn't compile, I mean serious debugging: breakpoints, figuring out value of variables at a given moment during run/compile, stepping, examining a stack (is there one?), etc. All the things that we take for granted when debugging a "real" programming language. Since TeX is "turing-complete", I would like to be able to really see what is going on internally.   I'm guessing that a GUI interface for the debugger would be too much to ask, but if it exists, I'd love to hear about it.
If I took a picture of a windmill on the horizon — given that I know the sensor size and the focal length of the lens and other factors to do with the shot — could I calculate how far away an object is from the photographer? I was working on a problem in group theory, which asks about the automorphism group of a direct product of groups.  Okay, so I know that if $G,H$ are two groups whose orders are relatively prime, then $\operatorname{Aut}(G\times H)\cong\operatorname{Aut}(G)\times \operatorname{Aut}(H)$.  I also know that if $G,H$ are abelian and simple and isomorphic, then $\operatorname{Aut}(G\times H)\cong \operatorname{GL}_2(\mathbf{Z}_p)$ where $p$ is the order of $G$.  What I don't know is the following: what if $G,H$ are simple, yet not abelian? Let us for now focus on a more restricted case where $G\cong H$, in particular, $G=H$. What can we say about the group $\operatorname{Aut}(G\times H)$? (I heard something about this group being related to what is called "wreath product", which I do not know...)  Thanks bunch in advance! How to install Ubuntu using a smart phone as the installation media? How to get a dock on multiple monitors I have two monitors, but only one dock, which is somewhat annoying, so my question is, how do I get multiple docks? (One on each screen) How to create a circular shaped grid, similar to a manhole cover Which infinity is meant in limits? For example, when we write $\lim_{x\rightarrow \infty} f(x)$ - which infinity is meant and why? Countable? If uncountable - which and why? Will i be granted ESTA after getting rejected thrice on different nationality? I had applied for B1/B2 on my previous nationality but it got rejected Thrice.  I had planned on visiting my mother as she lives there. my last refusal was 3 years back. Now i have gained an European nationality and I am thinking of applying for ESTA so I can visit my mother. I have doubts if it will be accepted because I'll have to disclose about my previous refusals.  What should I do? because if I don't disclose my previous refusals I fear I might be rejected permanently. QT MySql connectivity using Windows XP, Qt Creator 4.5.2(windows 32 bit) How do I summon the Golem? Label PGFPlot table points with coordinates I would like to label each plotted point with its x and y coordinate. I am able to label with a single one using the MWE below. I must avoid manually labeling each point since my actual table size is larger and will be changed out regularly. \documentclass{standalone}   \usepackage{tikz} \usepackage{pgfplots} \pgfplotsset{compat=1.16}  \begin{document}  \pgfplotstableread[col sep=&amp;,row sep=\\]{  x     &amp;  y     \\  0.940 &amp;  0.992 \\  0.992 &amp; -0.940 \\ -0.992 &amp;  0.940 \\ -0.940 &amp; -0.992 \\ }\mytable  \begin{tikzpicture}  \begin{axis}     \addplot+[only marks, nodes near coords]table[x=x,y=y,meta=x]{\mytable}; \end{axis}  \end{tikzpicture} \end{document}  which creates the following figure  The desired output is for each point to be labelled with both coordinates. For specific example the top left point would be labelled with &quot;(0.94, 0.992)&quot;. Are the square roots of all non-perfect squares irrational? How to do a generalized linear model with multiple dependent variables in R? I have six dependent variables (count data) and several independent variables, I see that in a MMR the script goes like this:  my.model &lt;- lm(cbind(DV1,DV2,DV3,DV4,DV5,DV6) ~ IV1 + IV2 + ... + IVn)   But, since my data are counts, I want to use a generalized linear model and I tried this:  my.model &lt;- glm(cbind(DV1,DV2,DV3,DV4,DV5,DV6) ~ IV1 + IV2 + ... + IVn, family="poisson")   And appears this error message:  Error in glm.fit(x = c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  :    (subscript) logical subscript too long`   Can anyone explain me this error message or a way to solve my problem? Generating DLL assembly dynamically at run time Currently I have some code that is being generated dynamically.  In other words, a C# .cs file is created dynamically by the program, and the intention is to include this C# file in another project.  The challenge is that I would like to generate a .DLL file instead of generating a C# .cs file so that it could be referenced by any kind of .NET application (not only C#) and therefore be more useful. Background  I am a first year calculus student so I would prefer if answers remained in Layman's terms.  It is common knowledge and seems to me a mantra that I keep hearing over and over again to "not treat differentials/derivatives as fractions".  I am of course, in particular, referring to Leibniz notation.  However, aside from a quick response such as "oh, it's because its not a fraction but rather a type of operator", I never really got a full answer as to why we can't treat it as such. It just kind of sits at the edge of taboo in my mind where it sometimes gets used and sometimes doesn't.  Confusion is further compounded when a lot of things seem to just work out if we treat them just as fractions (e.g. u-substitution / related-rates)    Example     Air is being pumped into a balloon at a rate of $100cm^3/s$. We want the rate of change of radius when the radius is at $25cm$.   $$\text{we are given}\ \frac{dv}{dt}=100cm^3/s$$ $$\text{we want}\ \frac{dr}{dt}\ \text{when}\ r=25cm$$ Thus we will solve this by using the relation $v=\frac{4}{3}\pi r^3$ $$\frac{dv}{dt}=\frac{dv}{dr}\frac{dr}{dt}$$ $$\frac{dv}{dt}\frac{dr}{dv}=\frac{dr}{dt}$$ $$100\frac{1}{4\pi r^2}=\frac{1}{25\pi}$$ So the answer is $\frac{dr}{dt}=\frac{1}{25\pi}$ when $r=25cm$  *Note the manipulation of derivatives just as if they were common fractions using algebra.    Question  When exactly can I treat differentials/derivatives as fractions and when can I not?   Please keep in mind that at the end of the day, I am a first year college student. An answer that is easy to understand is preferred over one that is more mathematically rigorous but less friendly to a beginner such as me. I want to enable GPU rendering, but there is no option in User Preferences > System:    Why is this? How can I get cycles to render using my GPU? Here is the top output which I gathered:    I noticed that top is showing vlc's cpu usage is > 100%.  Can anybody please justify me why it is showing like this, whether this is a bug in top application or something else. GDBus.Error when trying to enable second screen I connected an external screen to my laptop, but my desktop didn't automagically extend to it, as it used to do about a year ago (older version of Ubuntu).  I went to the display settings. The screen was listed there (correct resolution, brand name, etc.). But when i put the switch on 'on' and click 'apply', i get this error:   Failed to apply configuration: %s  GDBus.Error:org.freedesktop.DBus.Error.UnknownMethod: No such interface `org.gnome.SettingsDaemon.XRANDR_2' on object at path /org/gnome/SettingsDaemon/XRANDR   It's been like this for a few months now, but i'd really like to be able to use an external screen/beamer again.  Any ideas?  Thanks. Is $\mathbb{Z}[x]/(x^2+1)$  isomorphic to  $\mathbb{Z}[i]$?   My attempt is that try to define a mapping $g$ from $\mathbb{Z}[x]$ to $\mathbb{Z}[i]$ by $g(f(x))= f(i)$, for $f(x)\in\mathbb{Z}[x]$. If it is possible then $\ker g$ is $(x^2+1)$? Am I on the right track? Please Help. Should we write “allow” or ”allows” in “require that the class allow[s]”? Can you explain to me whether I should or should not use ‑s at the end of the  verb allow in this sentence?     This function requires that the container class allow[s] random access   The sentence in the book I read wrote allow without ‑s. However, the container class is a third-person singular, so the ‑s must be required. Is that right? Breakthrough at the Large Hadron Collider -Leptoquarks skepticism?