Mathematical induction $3$ and $5$ cent coins
$3$ and $5 $cent coins
Posting and answering questions you have already found the answer to Sometimes you spend a day or more solving a technical problem, and when you finally solve it, you may want to post it on your blog (if you have one) to share with the community.  However, most developers don't have a lot of time to update their blogs, and that is why personal blogs are not ranked very highly on Google/Bing.  So even if you describe a brilliant idea, chances are that, except a few of your friends, nobody will ever find it. This is a common reason why some developers start their blogs and soon give up; it is not interesting to write for the community if you have no feedback on your posts.  Unlike personal blogs, Stack Exchange sites are highly ranked in Google (I was amazed when I saw that a question asked on SO just an hour ago appeared 3rd in my Google search!), so if your question and answer are valuable for others, they will be rated highly and will be visible for other developers.  With this in mind, is it good practice to post a question and answer it yourself immediately when you want to share an idea with the community? How to do multiple lines within braces? Does this seemingly elementary question require König's theorem? TikZ: multiple decorations on a path. Suppose that $X$ is a topological space equipped with a delta-complex structure. Prove that $X$ is Hausdorff.  I can actually "see" why it should be Hausdorff after the hint from Hatcher asks to "think of a delta-complex X as a quotient space of a collection of disjoint simplices".  I would appreciate if somebody can write out a full formal argument showing this proof. I'm trying to start from the definition of Hausdorff but I don't know how to carry on. Number of ways to write n as a sum of k nonnegative integers How many ways can I write a positive integer $n$ as a sum of $k$ nonnegative integers up to commutativity?  For example, I can write $4$ as $0+0+4$, $0+1+3$, $0+2+2$, and $1+1+2$.    I know how to find the number of noncommutative ways to form the sum: Imagine a line of $n+k-1$ positions, where each position can contain either a cat or a divider. If you have $n$ (nameless) cats and $k-1$ dividers, you can split the cats in to $k$ groups by choosing positions for the dividers: $\binom{n+k-1}{k-1}$. The size of each group of cats corresponds to one of the nonnegative integers in the sum. How do I run wireshark, with root-privileges? A standard installation of Wireshark doesn't give the program permission to access the network interface.   I suppose I have to run the program with sudo, but do not know how to add it to the icon - if that's the way to do it. 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! accounting for LED resistance What is the measure of the set of numbers in $[0,1]$ whose decimal expansions do not contain $5$? Why do we have an elementary charge $e$ in physics but no elementary mass? Is an elementary mass ruled out by experiment or is an elementary mass forbidden by some theoretical reason? I recently decided to watch Doctor Who, and started viewing the 2005 version. I have the first two episodes from the first season, and I can't help but wonder what is the etymology of the name "Doctor Who"? And why does the protagonist call himself "the Doctor" (or is it "the doctor")? Sum of a power series $n x^n$ I would like to know: How come that  $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$  Why isn't it infinity? NSString question - rangeOfString method Referencing to "above" or "below"? What is glue stretching? I had a question about dynamic vspaces () and there was a hint I should use a stretchable glue. Searching with Google didn't give me a helpful answer.  I would be glad, if someone can explain it and give me a simple example to understand how it works. I need to know a word which means:     "negligence shown by a person in performing a duty, or a responsibility, or any given task."   or,      "A person who is negligent in performing their duties/responsibilities."   One possible choice is the use of word "delinquent". Is there any other word?  Please provide the usage of suggested word in a sentence also. All matrices which commute with all $2\times 2$ matrices I would like to find all matrices which commute with all $2\times2$ matrices.  I started solving problem in this way:    1) I have this matrix $A$ with real numbers: $$A=\left[\begin{array}{cc}a &amp;b\\c &amp;d\end{array}\right]$$ 2) Matrix which commute with matrix $A$ is matrix $B$: $$B = \left[\begin{array}{cc}e&amp; f\\ g &amp;h\end{array}\right]$$ 3) When i solve the equation $AB=BA$, I get this: $$\left[\begin{array}{cc}ae+bg&amp;af+bh\\ce+dg&amp;cf+dh\end{array}\right]=\left[\begin{array}{cc}ea+cf&amp;eb+df\\ga+ch&amp;gb+dh\end{array}\right]$$  How I can get the general look of wanted matrix? Can the Flash do things like build muscle faster than normal (do his muscles grow right after a work-out or meal)? Can he heal faster than normal? Does he have a faster immune system to stop all illness or at least make the flu last 2 minutes or something along those lines? Setting print layout to portrait using QGIS 3?