For example I need to specify a term x subscript a subscript b.  For a x subscript a, I can write it as x_{a}. However, x_{a}_{b} is not supported. How do I do that?
What's the best way to write subsequences?  $x_n_i$ gives an error, while $x_{n_{i}}$ works, but it just looks like $x_{ni}$ (the i isn't lowered enough below the n)...
How to reference sections, equations, and figures in a separate document (e.g., supplemental information)? I have a main document (manuscript) and a supplemental document.   I wonder if there is a way that I can "link" the two (make them recognize each other) so that section, equation, and figure references across documents are recognized? Isomorphisms preserving integral domains and fields Let $R \simeq S$ be isomorphic commutative rings with unity. Prove the following:  a). If $R$ is an integral domain then $S$ is an integral domain.  For this, I said that If I let $f: R \to S$ be an isomorphism and $(R, + , \circ), (S, \oplus, \ast)$, then let $x, y\in R$,  such that $x \circ y = 0_R$, because $f$ is an isomorphism, $f(x \circ y) = f(x) \ast f(y) = f(0_R)$. If I let $R$ be an integral domain then, as per the definition $R$ is commutative, has unity, and no zero divisors.   So $f(x \circ y) = f(x) \ast f(y) = f(0_R)$ where we have either $f(x \circ y) = f(0_R) \ast f(y) = f(0_R)$ or $f(x \circ y) = f(x) \ast f(0_R) = f(0_R)$.   Since $f$ is an isomomorphism, let $s = f(x), t = f(y)| s,t \in S$. Then $s \ast t = f(0_R)$. By definition, $S$ also has unity so $0_S$ exists. If we map $0_R \to 0_S$. Then either $s = 0_S$ or $t = 0_S$.   So then, since $S$ has unity, is commutative, and has a no zero divisors as we showed above, it is also an Integral domain.  I'm not sure I have this correctly.  b). If $R$ is a field then $S$ is a field. I don't really understand how to do this one. Since $S$ is an integral domain, it has a $0_S$. So I need to show that $S$ distributes $\ast$ over $\oplus$.  Any help with these is appreciated Choosing Geographical Coordinate System and Projected Coordinate System for India? I want to create spatial data for entire India.  Usually I use WGS84 as GCS and UTM zones for PCS in the places in India, but when I have to create the spatial data for entire India, I get confused which geographical coordinate systems and projected coordinate system will be suitable.  What are suitable ones? How to monitor process/program execution in windows? We are trying to develop a small application that can monitor the programs/processes that are executing in a windows machine.  If the program/process is not supposed to run, it should be blocked. It works similar to an antivirus.  This is the basic idea.  I want to know the ways to hook into the OS to get notified about every single program/process trying to run in the machine. Invariant Subspace of Two Linear Involutions I'd love some help with this practice qualifier problem:  If $A$ and $B$ are two linear operators on a finite dimensional complex vector space $V$ such that $A^2=B^2=I$ then show that $V$ has a one or two dimensional subspace invariant under $A$ and $B$.    Thanks! Is time continuous or discrete? I was coding a physics simulation, and noticed that I was using discrete time. That is, there was an update mechanism advancing the simulation for a fixed amount of time repeatedly, emulating a changing system.  I though that was interesting, and now believe the real world must behave just like my program does. It is actually advancing forward in tiny but discrete time intervals? I received a table template, which I'm intended to generate dynamically with data using Java. But when the table becomes to large, the compilation with LaTeX fails. The specific about this template is, that it stretches the table width into the maximal page width and has same column widths for all data columns (except the first column).  Here is a minimal version of the full template, that still works fine:  \documentclass[final,authoryear,3p,twocolumn]{elsarticle}  \usepackage{amssymb,amsmath} \usepackage{calc} \usepackage{array} \newcolumntype{L}[1]{&gt;{\raggedright\arraybackslash}p{#1}} \newcolumntype{R}[1]{&gt;{\raggedleft\arraybackslash}p{#1}} \usepackage{eurosym} \usepackage{microtype} \usepackage{needspace} \usepackage{fp}  \usepackage{graphicx} \DeclareGraphicsExtensions{.pdf} \input{pgfexternal}  \newcommand\tabt{\rule{0pt}{2.6ex}} \newcommand\tabb{\rule[-1.2ex]{0pt}{0pt}}  \begin{document}  \begin{table*}[t]     \footnotesize     \begin{tabular*}{\linewidth}{@{}L{2em}@{}*9{R{(\linewidth-2em)/9}@{}}}\hline\tabt $i$ &amp; $1$ &amp; $2$ &amp; $3$ &amp; $4$ &amp; $5$ &amp; $6$ &amp; $7$ &amp; $8$ &amp; $9$\tabb\tabularnewline\hline\tabt $1$ &amp;  &amp; $48.87$ &amp; $0.01$ &amp; $0.05$ &amp; $1.23$ &amp; $0.00$ &amp; $0.20$ &amp; $5.67$ &amp; $2.35$\tabularnewline     \end{tabular*} \end{table*}  \end{document}   But when extending this example by another column, so that the table would look like the following example, it fails:  \begin{table*}[t]     \footnotesize     \begin{tabular*}{\linewidth}{@{}L{2em}@{}*10{R{(\linewidth-2em)/10}@{}}}\hline\tabt $i$ &amp; $1$ &amp; $2$ &amp; $3$ &amp; $4$ &amp; $5$ &amp; $6$ &amp; $7$ &amp; $8$ &amp; $9$ &amp; a\tabb\tabularnewline\hline\tabt $1$ &amp;  &amp; $48.87$ &amp; $0.01$ &amp; $0.05$ &amp; $1.23$ &amp; $0.00$ &amp; $0.20$ &amp; $5.67$ &amp; $2.35$ &amp; b\tabularnewline     \end{tabular*} \end{table*}   This fails, no matter how many columns I add. So far, it seems for me to be a problem with the division by 10. May it be, that LaTeX isn't able to handle that properly by default? How can I solve this problem and make even larger tables work using this template. Or is that to any other problem I don't see yet? Thanks for any help! What causes the 'Connection Refused' message? This is a  about Connection Refused    We see a lot of questions to the effect      When I try to connect to a system I get a message      Connection refused      Why is this ? Eclipse was working as good as anything on 14.04. I did a clean install of 16.04 and installed Eclipse. But it runs a Java program only once after which it just hangs during subsequent attempts before I remove and reinstall it. I have tried many things, since in the beginning I needed CDT as well. Right now, I just want to remove everything related to eclipse and do a fresh install (remove all the configs and dependencies). Please note that I used Eclipse installer to install Eclipse and the eclipse icon image is also not appearing (the default ubuntu ? is appearing). "I think him to be about 50" or "I think he is about 50"? I have two options. Which one is correct?      a) I think him to be about 50.   b) I think he is about 50.   If both are correct, should I avoid one or the other? Why should the first person pronoun 'I' always be capitalized? How to concatenate text from multiple rows into a single text string in SQL server? Consider a database table holding names, with three rows:  Peter Paul Mary   Is there an easy way to turn this into a single string of Peter, Paul, Mary? Accidentally hit "toggle full screen", How do I exit? I'm stuck in full screen in the node editor. Escape isn't working, and I have no menus and no idea how to leave except to quit Blender and reopen the program. Drawing on an image with TikZ Recently I have learned how to draw simple stuff with TikZ. I really like it. Now I wonder if it is possible to draw with TikZ on a image. Instead of pasting a image in Illustrator and adding some vectors to point something in the picture I would like to do it using TikZ. Is it possible?   For example, on a picture of a computer (PNG,etc) I would like to draw some arrows and labels saying "Keyboard", "Monitor", etc. Difference of the \dots* I came to wonder, what is the difference between the following?  \dotsc \dotso   As well as between:  \dotsb \dotsi \dotsm Normalizing the final weights vector in the upper bound on the Perceptron's convergence The convergence of the "simple"  says that:  $$k\leqslant \left ( \frac{R\left \| \bar{\theta} \right \|}{\gamma } \right )^{2}$$  where $k$ is the number of iterations (in which the weights are updated), $R$ is the maximum distance of a sample from the origin, $\bar{\theta}$ is the final weights vector, and  $\gamma$ is the smallest distance from $\bar{\theta}$ to a sample (= the margin of hyperplane).  Many books implicitly say that  $\left \| \bar{\theta} \right \|$ is equal to 1. But why do they normalize it ? How do you declare an interface in C++? How do I setup a class that represents an interface?  Is this just an abstract base class? I'm using JUnit 4.4 and Maven and I have a large number of long-running integration tests.  When it comes to parallelizing test suites there are a few solutions that allow me to run each test method in a single test-class in parallel. But all of these require that I change the tests in one way or another.  I really think it would be a much cleaner solution to run X different test classes in X threads in parallel. I have hundreds of tests so I don't really care about threading individual test-classes.  Is there any way to do this? I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their double duals; they are isomorphic to their duals as well, but not canonically."  I've finally sat down to work through this, and -  Okay, yes, it is easy to see that the "canonical isomorphism" from $V$ to $V^{**}$ is a functor that has a natural isomorphism (in the sense of category theory) to the identity functor.  Also, I see that there is no way that the functor $V\mapsto V^*$ could have a natural isomorphism to the identity functor, because it is contravariant whereas the identity functor is covariant. My question amounts to:     Is contravariance the whole problem?   To elaborate:   I was initially disappointed by the realization that the definition of natural isomorphism doesn't apply to a pair of functors one of which is covariant and the other contravariant, because I was hoping that the lack of a canonical isomorphism $V\rightarrow V^*$ would feel more like a theorem as opposed to an artifact of the inapplicability of a definition.  Then I tried to create a definition of a natural transformation from a covariant functor $F:\mathscr{A}\rightarrow\mathscr{B}$ to a contravariant functor $G:\mathscr{A}\rightarrow\mathscr{B}$. It seems to me that this definition should be that all objects $A\in\mathscr{A}$ get a morphism $m_A:F(A)\rightarrow G(A)$ such that for all morphisms $f:A\rightarrow A'$ of $\mathscr{A}$, the following diagram (in $\mathscr{B}$) commutes:  $$\require{AMScd}\begin{CD} F(A) @&gt;m_A&gt;&gt; G(A)\\ @VF(f)VV @AAG(f)A\\ F(A') @&gt;&gt;m_{A'}&gt; G(A') \end{CD}$$  This is much more stringent a demand on the $m_A$ than the typical definition of a natural transformation.  Indeed, it is asking that $m_A=G(f)\circ m_{A'}\circ F(f)$, regardless of how $f$ or $A'$ may vary. Taking $\mathscr{A}=\mathscr{B}=\text{f.d.Vec}_k$, $F$ the identity functor and $G$ the dualizing functor, it is clear that this definition can never be satisfied unless $m_V$ is the zero map for all $V\in\text{f.d.Vec}_k$ (because take $f$ to be the zero map). In particular, it cannot be satisfied if $m_V$ is required to be an isomorphism.     Is this the right way to understand (categorically) why there is no natural isomorphism $V\rightarrow V^*$?   As an aside, are there any interesting cases of some kind of analog (the above definition or another) of natural transformations from covariant to contravariant functors?  Note: I have read a number of math.SE answers regarding why $V^*$ is not naturally isomorphic to $V$. None that I have found are addressed to what I'm asking here, which is about how categories make the question and answer precise. ( was closest.) Hence my question here. Meaning of Conditional sentences with 'should' and 'happen to'