Show that $\mathrm{min}(\frac{a}{b},\frac{a'}{b'}) \leq \frac{a+a'}{b+b'} \leq \mathrm{max}(\frac{a}{b},\frac{a'}{b'})$
shorter proof of generalized mediant inequality?
If Chrome is already open, it opens new sites (e.g. from Windows shortcuts, application shortcuts, or from the Run dialog) in new tabs of the existing instance.   I'd prefer to open new sites in a separate window (a new instance of Chrome).   I don't see that in the options - is it possible? Let $p$ be a prime number . The order of a $p$-Sylow subgroup of the group $GL_{50}(F_p)$ of invertible $50\times50$ matrices with entries from the finite field $F_p$,equals which of the following    1.$p^{50}$ 2.$p^{125}$ 3.$p^{1250}$ 4.$p^{1225}$    I think order of this  group is $p^{2500}$  .A  sylow p-subgroup of order $P^k$ divides  order of the group but $p^{k+1}$  does not. I confused here how to apply these things . For quite some time now I have been asking myself how the world of Game of Thrones is shaped and if the people living in that world know the shape.   In questions here on SciFi.Stackexchange concerning the unnatural seasons or the definition of a year in Westeros, it is often  that the Westerosi can measure the duration of a revolution of their planet around the sun. But that only works if they know that they actually live on a planet.   In  answer it was said that people have been travelling in ships around the world. However, I cannot remember any passage in the books supporting this. Actually I only remember evidence for the opposite:    During the  on the Iron Islands, there is a man called  who promises the Ironborn treasures from lands far to the west, if they elect him as king. However, the POV character of that chapter (I think it was Victarion Greyjoy) does not believe him and sees the madness in Gylbert's eyes - as if speaking of travelling west is crazy.  Later when he leads the attack against the Shield Islands, the Iron Fleet travels out of sight of the coast line. This manoeuvre is considered very dangerous and risky. If this is already dangerous, traveling even further west appears close to suicide.   In addition, there was a  who sailed to the far west and never returned.   establishes that the world is not a hollow world, but is not conclusive as to whether it is a planet.  Is there any evidence in the books that the World of Ice and Fire is a planet? Was it established that the Westerosi know the nature of their world? How to do one-class text classification? I have to deal with a text classification problem. A web crawler crawls webpages of a certain domain and for each webpage I want to find out whether it belongs to only one specific class or not. That is, if I call this class Positive, each crawled webpage belongs either to class Positive or to class Non-Positive.  I already have a large training set of webpages for class Positive. But how to create a training set for class Non-Positive which is as representative as possible? I mean, I could basically use each and everything for that class. Can I just collect some arbitrary pages that definitely do not belong to class Positive? I'm sure the performance of a text classification algorithm (I prefer to make use of a Naive Bayes algorithm) highly depends on which webpages I choose for class Non-Positive.  So what shall I do? Can somebody please give me an advice? Thank you very much! Updating list definition using features I have a feature that deploys content types, a list definition and template that uses these content types and finally a list instance. This feature is already in production.  Now we have a change request to add new views to this existing list. So I make the changes to schema.xml and upgrade the solution. But alas, the new views don't show up. If I create a new site, the list instance is created with the new views. I tried deactivating/reactivating, iisreset but still no success. Is there anything I'm missing? Clarify an idiom in a political text Please carefully read the text below:     On 24 November, 1993, a meeting of Leftist intellectuals occurred in   London under the auspices of the Institute for Public Policy   Research (IPPR), which is a Labour-leaning think-tank.  A short document   was circulated in advance of the said meeting to clarify its   purpose.        Among other things, the document declared that the task of the   IPPR was: "To do what the Right did in the seventies, namely to break   through the prevailing parameters of debate and offer a new   perspective on contemporary British politics."    The explanatory document also said, "Our concern is not to engage in a   philosophical debate about foundations of socialism."      If this meant that those foundations were not the appropriate thing to   talk about at the 24 November meeting, then that might have been   right; not every thing has to be discussed at every meeting.  But if   what was meant was that discussion of philosophical foundations is not   what the Left now needs, then I disagree, and if that indeed is what   was meant, then it is curious that the breakthrough by the Right should   have been invoked as an achievement for the left to emulate, for if   there is a lesson for the Left in the Right's breakthrough, it is that   the Left must repossess itself of its traditional foundations on   pain of continuing along its present, politically feeble, reactive   course.  If the Left turns its back on its foundations, it will be   unable to make statements that are truly its own.   Since I've heard only one definition of the part identified in bold, I am at a dead end.  As you know, and according to Oxford's dictionary, the verb "repossess" means: To take back property or goods from SB who has arranged to buy them and can not pay.  Here, the object of verb is a thing not a person, but in the text, first,itself comes before of and its traditional foundations. As a result, I thought that this phrase might be an idiom and so has a different meaning with the original verb.  Please explain the meaning of this phrase. GeoServer cutting off symbols close to tile edges? Which of these two is more correct and why?   The sun shines on my little tree. The sun shines upon my little tree.   From what I've read in dictionaries both are acceptable, however I would like to know if there are subtleties that might arise due to, for example, the context. I am new to the glmnet package, and I am still unsure of how to interpret the results. Could anyone please help me read the following trace plot?  The graph was obtaining by running the following:  library(glmnet) return &lt;- matrix(ret.ff.zoo[which(index(ret.ff.zoo)==beta.df$date[2]), ]) data   &lt;- matrix(unlist(beta.df[which(beta.df$date==beta.df$date[2]), ][ ,-1]),                   ncol=num.factors) model  &lt;- cv.glmnet(data, return, standardize=TRUE)  op &lt;- par(mfrow=c(1, 2)) plot(model$glmnet.fit, "norm",   label=TRUE) plot(model$glmnet.fit, "lambda", label=TRUE) par(op) Windows service stops automatically Showing OSM source acknowledgment on the map in QGIS? I am using OpenStreetMap as a base layer via XYZ tiles in QGIS 3.01.  I can't seem to find a way of having source acknowledgement appear on the lower right corner of the map.  Can somebody point me in the right direction? nodes near coords with logarithmic axis: Wrong values I created a horizontal bar plot to compare benchmark times. Since one of the tested components is really slow, I need a logarithmic y axis for the other values to be distinguishable. Combined with "nodes near coords" this results in the calculated values showing, instead of the original ones. Is there any way to fix this?  What it looks like:   What the values(!) should look like:   Clearly, I'm overlooking something terribly obvious. I'm just a beginner at PGFPlots, so please don't kill me. :)  MWE:  \documentclass[fontsize=12pt]{scrartcl}  \usepackage{pgfplots} \pgfplotsset{compat=1.8}  \begin{document}   \begin{tikzpicture}   \begin{axis}[      width=12.5cm,     xbar=0.1cm,     xmin=0,     y=0.6cm,     bar width=0.4cm,     xlabel={Time (ms)},     xmode=log,     major x tick style = transparent,     major y tick style = transparent,     symbolic y coords={       {SpreadsheetGear},       {Aspose.Cells},       {NPOI},       {C1.Excel},       {Spire.XLS},       {GemBox.Spreadsheet},       {SmartXLS},     },     ytick=data,     y dir = reverse,      nodes near coords,     nodes near coords align={horizontal},   ]    \addplot coordinates {       (3078.6,{Aspose.Cells})       (10059.2,{C1.Excel})       (28794.8,{GemBox.Spreadsheet})       (7437079.8,{SmartXLS})       (19975.2,{Spire.XLS})       (573.2,{SpreadsheetGear})       (10025.2,{NPOI})};   \end{axis}   \end{tikzpicture} \end{document} Is there a way to send a thanks message to Stack Overflow or any of the individual Stack Exchange sites, aside from sending an email? I want to thank Stack Overflow for being an awesome live QA site. It helps people a lot. Referencing equation counter value in custom float caption I have used the float package to create an equation float, eqfloat, so I can add captions to equations. Within the eqfloat I used the equation environment to typeset the math.  I want to keep the right hand arabic numbering within parentheses that is added by the equation environment but I also would like the eqfloat caption below the equation to use the same number as the equation environment.  The following code achieves this by making the eqfloat use the same counter as the equation environment and then manually adjusting the equation numbering within each eqfloat.   Is there a better way to make the eqfloat caption reference the current value of the equation environment counter?  Is it possible to prevent the custom float environment from increasing the counter and only to reference the current value of the equation counter?  \documentclass[a4paper]{article}  \usepackage[margin=25pt, labelfont=bf]{caption} \usepackage{amsmath, amsfonts, pifont, float, color, url} \usepackage[pdftex, plainpages=false, pdfpagelabels, bookmarks=true]{hyperref}  \newfloat{eqfloat}{h}{eqflts} \floatname{eqfloat}{Equation} \makeatletter \let\c@eqfloat\c@equation \makeatother  \begin{document}     ...     \begin{eqfloat}         \begin{equation}             E = mc^2         \end{equation}         \addtocounter{equation}{-1}         \caption{Equation caption here.}         \label{eq:example}     \end{eqfloat}  \end{document} Why does the product topology allow proper subsets for only finitely many elements? Schrödingers MySQL table: exists, yet it does not How to ignore deprecation warnings in Python Reason for strange "canyons" of dust on the cooling platform I took apart my cooling platform Something like this :  And I found something like this inside :       I wonder why around the hollows are the "canyon like" structures.  PS: I also drop this platform on the floor, maybe this could be a reason ?   EDIT :  Everybody seems (except Brandon Enright answer) to ignore the word "dust" in this question. Those structures are not in the plastic itself. It is dust ON the plastic.  Proof :     Completely wiped out with my finger : I get how a normal fork bomb works, but I don't really understand why the &amp; at the end of the common bash fork bomb is required and why these scripts behave differently:  :(){ (:) | (:) }; :   and   :(){ : | :&amp; }; :   The former causes a cpu usage spike before throwing me back to the login screen. The latter instead just causes my system to freeze up, forcing me to hard reboot. Why is that? Both continually create new processes, so why does the system behave differently?   Both of the scripts also behave differently from  :(){ : | : }; :   which doesn't cause any problems at all, even though I would have expected them to be alike. The bash manual page states that the commands in a pipeline are already executed in a subshell, so I'm led to believe that : | : should already suffice. I belive &amp; should just run the pipeline in a new subshell, but why does that change so much?  Edit: Using htop and limiting the amount of processes, I was able to see that the first variant creates an actual tree of processes, the second variant creates all the processes on the same level and the last variant doesn't seem to create any processes at all. This confuses me even more, but maybe it helps somehow? I recently got the book "selected problems in real analysis", and I'm stuck solving the very first problem  $(u_n)$ is a binary sequence iff it only contains $0$ and $1$ in the sequence  Let $A$ be the set of all binary sequences  I have to prove that $A$ and ${2}^{\mathbb{N}}$ have the same cardinality, that is to say there exists a 1-1 function from one set to another  I've thought about maybe considering integers as base-2 numbers  Thanks for your help