So I'm a beginner to graph theory and really unsure how to approach this. I was thinking by counting Prufer codes, in which case you would want the second element in the Prufer code to be 3. So, it would be the number of Prufer codes where one of the labels are fixed at 3, so $n^{(n-1)-2}$ labeled trees?
How many labeled trees which have one vertex of degree 4 can we have? A tree with $10$ vertices has $9$ edges, so the sum of the degrees of the vertices is $2\cdot 9=18$. If there is one vertex of degree $4$, the sum of the degrees of the other $9$ vertices must be $14$. I don't know what to do next. Should I count all possible vertex degrees?
For distinct prime numbers $p_1,...,p_n$, what is the Galois group of $(x^2-p_1)\cdots(x^2-p_n)$ over $\mathbb{Q}$?     This problem appears to be quite common, however my understanding of Galois theory is quite poor, and I have no idea how to do this problem. How do I preview tag wiki exerpt? How do I put a link there? I've edited  to include a link into exerpt. But it shows Markdown source of the link instead of actual link.   How do I preview the exerpt while editing it? Where to find syntax and available Markdown features in exerpts? How do I include the link properly? Finding $ \int^1_0 \frac{\ln(1+x)}{x}dx$ There is supposed to be a clean solution to the integral below, maybe involving some symmetry $$ \int^1_0 \frac{\ln(1+x)}{x}dx$$ I have tried integration by parts as followed: $\ln(x+1)=u$ ,$\frac{1}{x+1} dx = du$ and also $\frac{1}{x}dx=dv$, $\ln(x)=v$. Then, the integral becomes $$\int^1_0 \frac{\ln(1+x)}{x}dx=- \int^1_0 \frac{\ln x}{1+x}dx$$ which does not make this easier. I have also tried using the identity $\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$. So let $I = \int^1_0 \frac{\ln(1+x)}{x}dx$ and also $I = \int^1_0 \frac{\ln(2-x)}{1-x}  $. Then $$2I= \int^1_0 \ln(1+x)+\ln(2-x)$$ which is not any easier, either. Any ideas? :) Why does kinetic energy increase quadratically, not linearly, with speed? As  says:     [...] the kinetic energy of a non-rotating object of mass $m$ traveling at a speed $v$ is $\frac{1}{2}mv^2$.   Why does this not increase linearly with speed? Why does it take so much more energy to go from $1\ \mathrm{m/s}$ to $2\ \mathrm{m/s}$ than it does to go from $0\ \mathrm{m/s}$ to $1\ \mathrm{m/s}$?  My intuition is wrong here, please help it out! Why is 8 so special? I have been reading about multi-dimensional numbers, and found out that it's been proven that the Octonions are the composition algebra of the largest dimension. I was wondering why, despite having infinitely many different dimensions of numbers, the only composition algebras are of 1, 2, 4, and 8 dimensions. What's so special about 8? I would like to simulate the behaviour of a zener diode in LTspice. Consider the following schematic         &ndash; Schematic created using   I'm interested in simulating its performance when Vcc changes from the rated value (say 12V), for example when Vcc varies from 11 to 15 V.  Now, I know how to simulate this behaviour in LTspice for a generic diode or even for a particular diode which is available in the model library, but when it comes to using a custom diode, I have no idea on how to specify the diode parameters.  Looking into the documentation I was able to come up with a setup that is the correct one, that is, I can specify a custom diode model but now I'm having problems in understanding where to find all the parameters since the datasheet of my diode of choice does not provide all of them. In the LTspice model shown below I've picked a 4.7 V zener diode that was available just to have a base model to edit (and a 1k resistor just for reference).    The diode I'd like to use is a 3V . How can I simulate, as realistically as possible, the behaviour of this diode given the few parameters available in the datasheet? Writing function definition in header files in C++ I have a class which has many small functions. By small functions, I mean functions that doesn't do any processing but just return a literal value. Something like:  string Foo::method() const{     return "A"; }   I have created a header file "Foo.h" and source file "Foo.cpp". But since the function is very small, I am thinking about putting it in the header file itself. I have the following questions:   Is there any performance or other issues if I put these function definition in header file? I will have many functions like this. My understanding is when the compilation is done, compiler will expand the header file and place it where it is included. Is that correct? Reporting Mann-Whitney U-Test without homogenity of variance The Mann-Whitney test requires homogeneity of variance if a median difference is suppossed to be statistically significant.  In case homogenity of variance is not met, but the test is significant:  Which aspects of the test can I report? Embed .mov file via 'Add Media' not working I know since WP 3.6 we've been able to upload and embed various file formats and that the 'Add Media' button effectively implements the correct media embed shortcode for us.  I'm having a problem in WP4.0 using 'Add Media' to embed quicktime (.mov) files. The embed works fine for the other supported video MIME types but I'm not offered 'Embed Media Player' when I try adding a .mov file to a post.  See screen grabs/explanation:   See how .mov embed has reverted to a simple hyperlink (recorders) at   Does anyone have any suggestions as to why this is, or to a possible fix so I can embed .mov as easily as .mp4 - seems to be the preferred format of the client. For the moment we're having to manually add the [video] shortcode ourselves in the text editor.  How do I check that embed is supported/enabled for .mov? What is a referee in the manuscript submission process? Please submit, with the manuscript, the names, addresses (including   countries), and e-mail addresses of five potential referees who are   outside the authors' institutions and must not have conflict of   interest with the authors.   Do you think I should name 5 researchers from a similar area that are not affiliated with my university and co-authors? Is this what referees are? I am giving a presentation and therefore I am using the  class. I have included some figures and it works, but there is no number after "Figure".  My code:  \documentclass{beamer}  \usepackage{graphicx} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage[english]{babel} \usepackage{listings} \usepackage{xcolor} \usepackage{eso-pic} \usepackage{mathrsfs} \usepackage{url} \usepackage{amssymb} \usepackage{amsmath} \usepackage{multirow} \usepackage{hyperref} \usepackage{booktabs} \usepackage{wallpaper} \usepackage{bbm} \usepackage{cooltooltips} \usepackage{eso-pic} \usepackage{everyshi} \usepackage{calc} \usepackage{graphicx} \usepackage{ifthen} \usepackage{graphicx} \usepackage{wrapfig} \usepackage{animate} \usepackage[singlespacing]{setspace} \usepackage[absolute,overlay]{textpos} \usepackage{enumerate} \usepackage{hyperref} \newenvironment{changemargin}[2]{%   \begin{list}{}{%     \setlength{\topsep}{0pt}%     \setlength{\leftmargin}{#1}%     \setlength{\rightmargin}{#2}%     \setlength{\listparindent}{\parindent}%     \setlength{\itemindent}{\parindent}%     \setlength{\parsep}{\parskip}%   }%   \item[]}{\end{list}}  \setbeamercolor{frametitle}{fg=red,bg=white}  \hypersetup{pdfpagemode=FullScreen} \renewcommand*{\UrlFont}{\normalfont}  \usetheme{Goettingen} \useinnertheme{circles} \beamertemplatenavigationsymbolsempty  \setbeamerfont{page number in head/foot}{size=\large} \setbeamertemplate{footline}[frame number]   %%% \begin{document}  \frame[plain]{ \titlepage }  \setbeamertemplate{background canvas}[default]  \frame{ \frametitle{}  \begin{figure} \begin{center} \includegraphics[scale=0.2]{varweakness} \caption{VaR as a quantile} \end{center} \end{figure}  }   \end{document}   Which gives the following frame with the figure:    (I am not sure what is causing the problem, therefore I did not delete any package.)  How can I get a number, e.g. Figure 1:? I can not see how mathematically one can divide the numerator and denominator by theta. Can some one explain this in a different way. When you divide two fractions you multiply the first fraction by the reciprocal of the second. This is not the case here.     Divide numerator and denominator by $\theta$. ($\sin\theta$ also works.) $$\eqalign{\lim_{\theta\to0}\dfrac{\sin\theta}{8\theta+\tan\theta}&amp;= \lim_{\theta\to0}\dfrac{\dfrac{\sin\theta}\theta}{8+\dfrac{\sin\theta}\theta\cdot\dfrac1{\cos\theta}}\\&amp;=\dfrac{\lim\limits_{\theta\to0}\dfrac{\sin\theta}\theta}{8+\lim\limits_{\theta\to0}\dfrac{\sin\theta}\theta\lim\limits_{\theta\to0}\dfrac1{\cos\theta}}\\&amp;=\dfrac1{8+1\cdot1}\\&amp;=\dfrac19.}$$ How do you change from icon to list view in 20.4 List questions: Community Wiki? Converting GeoPDF to vector format like CAD or shapefile? I want to convert a GeoPDF that has vectors entity in it to a vector format and preserve the real coordinates of these vectors. The GeoPDF has been exported from ArcMap and has layers in it.  I have try GDAL, but it rasterize the PDF before exporting it. Which is not good for me. Also I don't want a solution that would convert this image back to vectors, too complicated for the data I have.  Is someone aware of any tool that can do that? The resulting format can be a CAD file or a shapefile or anything that is a vector format. Are Ubuntu packages security audited? Is there an "exists" function for jQuery? How can I check the existence of an element in jQuery?  The current code that I have is this:  if ($(selector).length &gt; 0) {     // Do something }   Is there a more elegant way to approach this? Perhaps a plugin or a function? What are point releases in LTS versions? An entire function whose real part is bounded must be constant. Prove that if $n \in \mathbb{N}$, then $\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$ where $d(n)$ is the divisor function.