I usually work with the normal article latex environment, with one column, and I usually introduce figures using the position given by [h] (with means "place the float here, i.e., approximately at the same point it occurs in the source text (however, not exactly at the spot)".  However, in multicols (in this case, in two columns), the only way I know to introduce graphics is by means of float package and it only works with the position given by [H] ("put exactly here"). The problem is that I have several graphics and using the position given by [H] there are lots of empty spaces that looks quite bad.  The question is if there is some way to use [h] position or something equivalent in the multicols package. Thanks in advance.
I'm trying to get images in my document in two different ways:   On top of the entire page, centered On top of a single column   I tried it with both twocols and multicols. I ruled out twocols because it forces a new page every time you turn it on, because I need the abstract to be wide and the rest of the cover page to be two columns.  Here are some examples of both points (screenshot from a downloaded PDF):   I want to achieve both, but somehow I can only solve the first point with:  \end{multicols}  \begin{figure}   \centering     \includegraphics[width=0.5\textwidth]{figure}   \caption{Dataflow pipeline of the edge detector}   \label{fig:workflowedge} \end{figure}  \begin{multicols}{2}   Is it awkward to turn off multicols, then turn it back on again?  For my second point, I've tried everything, but almost always, the image doesn't show up at all! The only way I can get it to show at all is by specifying the H placement specifier, but that places the image inside the text where I defined it, not on top of the column.  This is my best try, but it hides the image and breaks all references to ??:  \begin{figure}[t!]   \centering     \includegraphics[width=0.25\textwidth]{figure}   \caption{Dataflow pipeline of the edge detector}   \label{fig:workflowedge} \end{figure}   and I'm getting the warning:  Package multicol Warning: Floats and marginpars not allowed inside `multicols' environment!.   How can I make a figure a non-float?  I also tried the figure* environment, but that just creates an image centered on top of the page, like my first point.
When and why is the [featured] tag removed by the Community user? I've found that the Meta (moderator only) tag  is automatically  by the  user. So, I want to know when and why is the tag  is removed by this user?  Is there any mechanism for this removal or is there simply a specified time period after which the  is automatically removed by the Community user? Compactness of $\operatorname{Spec}(A)$ In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact.   Now because the basic open sets $X_f = \{\mathfrak{p} \in \operatorname{Spec} (A) : \{f\} \not\subseteq \mathfrak{p} \}$ form a basis for the Zariski Topology it suffices to consider the case when   $$X = \bigcup_{i \in I} X_{f_i}$$   where $I$ is some index set. Then taking the complement on both sides we get that   $$\emptyset = \bigcap_{i \in I} X_{f_i}^c$$  so there is no prime ideal $\mathfrak{p}$ of $A$ such that all the $f_i$'s are in $\mathfrak{p}$. Now from here I am able to show that the ideal generated by the $f_i$'s is the whole ring as follows. Since there is no prime ideal $\mathfrak{p}$ such that all the $f_i \in \mathfrak{p}$, it is clear that there is no $\mathfrak{p}$ such that $(f_i) \subseteq \mathfrak{p}$ for all $i \in I.$ Taking a sum over all the $i$ then gives $$\sum_{i \in I} (f_i) = (1).$$  Now here's the problem:          How do I show from here that there is an equation of the form $1 = \sum_{i \in J} f_ig_i,$ where $g_i \in A$ and $J$ some finite subset of $I$?       This part has been giving me a headache. I am not sure if the finiteness bit has to do with algebra, topology or the fact that we are dealing with prime ideals.  This is not a homework problem but rather for self-study.  $\textbf{Edit:}$ I have posted my answer below after the discussion with Dylan and Pierre. How can I break an align environment for a paragraph? I would like to have two displayed equations, broken by a short paragraph, but with aligned equal signs (as if using the align environment).  Is there some way to "break" the align environment temporarily? Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by.$  The required condition is $a^2&gt;8b^2$.I dont know how to prove it.I tried.  Let $(h,k)$ be the point from where tangents are drawn to the parabola $y^2=4ax$.Let $m_1,m_2$ be the slopes of the tangents.Equations of tangents are $y=m_1x+\frac{a}{m_1}$ and $y=m_2x+\frac{a}{m_2}$ and equations of normal to the parabola $x^2=4by$ is $y=m_1x-2bm_1-bm^3_1$ and $y=m_2x-2bm_2-bm^3_2$.  As $y=m_1x+\frac{a}{m_1}$ and $y=m_1x-2bm_1-bm^3_1$  are the same lines.Therefore,$\frac{a}{m_1}=-2bm_1-bm^3_1$  $bm^4+2bm^2+a=0$  And I am stuck.I cannot get the desired condition.Please help me. Will Ubuntu get patched against Meltdown and Spectre? The following is a canonical post that attempts to cover a broad number of Meltdown and Spectre questions.  It is being maintained by Thomas Ward who is in contact with the Security and Kernel teams on this matter for other reasons as well.   I recently read about the  vulnerabilities.  I was wondering, will Ubuntu be patching against the vulnerabilities?  And is there an ETA on a fix? If $\Gamma$ is satisfiable then $\Gamma$ is consistent We say that $\Gamma$ is consistent if for no $\phi$, then both $\Gamma\vdash \neg\phi$ and $\Gamma\vdash\phi$. We say that $\Gamma$ is satisfiable if there is a structure $M$ s.t. $\models_M\gamma[s]$ for alle $\gamma\in\Gamma$.  I want to show that if $\Gamma$ is satisfiable, then $\Gamma$ is consistent.  I'm thinking we assume that $\Gamma$ is not consistent, and then show that $\Gamma$ is not satisfiable. So if $\Gamma$ is not consistent then both $\Gamma\vdash \neg\phi$ and $\Gamma\vdash\phi$ for some $\phi$. But by soundness theorem then $\Gamma\models\neg\phi$ and $\Gamma\models\phi$, meaning for structure $M$, we have $\models_M\neg\phi[s]$ (whereas $\not\models_M\phi[s]$) and $\models_M\phi[s]$. So would this mean that $\Gamma$ is not satisfiable? I'm a little confused, because wouldn't this instead be a contradiction? Solve for $x$ in $6x^2-25x+12+\frac{25}{x}+\frac{6}{x^2}=0$ I simplified to get $6x^4-25x^3+12x^2+25x+6=0$  Or $6 (x^2+1)^2+25x(1-x^2)=0$  Then I substituted $z=x^2+1$, to get $6z^2+25\sqrt{(z-1)}(2-z)=0$  I can't find a way to proceed further. Moving a doorbell transformer Can a doorbell transformer that is on the outside of a wall be place inside the wall behind the drywall. I’m concerned about it getting too hot. Mine is in the kitchen by the ceiling lights Practical uses for the "internal" keyword in C# Could you please explain what the practical usage is for the internal keyword in C#?  I know that the internal modifier limits access to the current assembly, but when and in which circumstance should I use it? As a mod when should I vote to close content that is not flagged? I know mods are supposed to step in to quickly close seriously problematic posts and otherwise allow the community to vote to close a post.  But what about when there are four close votes on a question, and I personally feel the question should be closed but isn't worthy of mod super-mega closing? Should I cast the final vote like a normal user (even though it's technically binding) or should I sit back and wait  for the community?  For me the problem is complicated because on UX.SE we hover close to the point where there's not enough high-rep activity to quickly close posts and I was one of those high rep users that used to VtC.  Should I always treat my mod close vote as binding and reserve it for extreme situations, or should I treat my close vote as "normal" when it's the fifth and final vote anyway? How to inject keystrokes via a shell script? I'm trying to inject keystrokes into the input daemon so as to simulate typing from a Bash script. Is this possible, and if so, how can I make it happen? What is the strict aliasing rule? Possible to save a list template and maintain lookup relationships? My issue is that I have two lists that I would like to move with their data.  While I am able to save both as templates and then upload to the new site, none of the lookup columns in my parent table are populated with data now.  Can this be done? 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As for this what I  was:     To add an event listener to an element that a template doesn’t own,   call addEventListener directly: this.addEventListener()   What does salesforce mean by an element that a template doesn't own? Isn't window an element that the template doesn't own? Nautilus context menu for current directory when there's lots of files / "Open in Terminal" What is the difference between `grep`, `egrep`, and `fgrep`? Can anyone tell me the technical differences between grep, egrep, and fgrep and provide suitable examples?  When do I need to use grep over egrep and vice versa? How can I execute command on startup (rc.local alternative) on Ubuntu 16.10 I am setting up quota on my Linode server running Ubuntu 16.10, and I get the following error      Cannot stat() mounted device /dev/root: No such file or directory   So to fix this, I reached  for the fix which is done by adding   ln -s /dev/xvda /dev/root /etc/init.d/quota restart   to the /etc/rc.local. But Ubuntu 16.10 doesn't uses rc.local anymore instead uses . What is the alternative for rc.local, How can I run the above commands on startup?  Also I enabled the service using systemctl enable rc-local.service but it didn't work for me. Any lead would be appreciated. Let $R$ be a simple ring and $T$ be a non-zero ring. Let $f\colon R \rightarrow T$ be a ring homomorphism. Show $f$ is injective.  Proof: $\ker f \lhd R$, so $\ker f=R$ or $\ker f=\{0\}$. If $\ker f=R$ then $R/R \cong \{0\}$, but $T \neq \{0\}$, which is a contradiction. I can't see why does the last part means contradiction.   Thanks for any assistance!