dimension of a direct sum is the sum of the separate dimensions
Proof that $V = \sum_k U_k$ and $\dim V=\sum_k \dim U_k$ implies $V=\oplus_k U_k$
How do websites block selecting text and how do I unblock that? There's a website () that somehow blocks selecting text. Also it blocks Ctrl+A to select everything, there is also no “Copy” in the popup menu.  What I have tried:  Some sites use JavaScript to block selection. So I disabled all JavaScript sources in no-script addon in Firefox. I tried the same site in Google Chrome with the Script Safe addon. Made sure that everything is blocked, still can’t select text. Mousepointer remains an arrow and not a text cursor also on the whole site.  Used about:config to disable JavaScript completely in Firefox. Still no effect.  I read that some sites use DIVs with the style display:block so I used Inspect to examine the styles of the site. There is not one mention of "block" on the whole website, not in embedded CSS nor in object style=-attributes.  The text is not an image or flash or some HTML5 canvas ect. Everything is within DIV and P tags but no style was found that might block the text selection.  How can the website still block any selection of text? Not to mention why browsers support such behaviours at all. I just want to use a selected word or sentence and want to search google for it using right mouse click. It’s pretty annoying when a website does such things and forces me to type technical terms and names into google by hand. It disturbs my workflow. How do I connect Android devices to a WiFi AP and mobile data network simultaneously? When I turn WiFi on, my 3G connection shuts down, which is fine in most cases. However, I need to connect to a LAN at work that is not connected to the internet and I'd prefer not to lose my data connection when I connect to the work intranet. How can I keep both connections active? Ideally, I could limit this desired behavior to this one SSID because I want it to continue to operate with the default behavior for all other WiFi connections.  If that can't be done, prefer a solution that is easy to toggle on and off. I'm using a Samsung Galaxy Nexus (Verizon Wireless) on CyanogenMod 9. Proving that the triangle inequality holds for a metric on $\mathbb{C}$ Show that $(X,d)$ is a metric space where $X =\Bbb C $ and the distance function is defined as:   $$d(x,y) = \frac {2|x-y|}{\sqrt {1+|x|^2} +  \sqrt {1 + |y|^2}}, \text{ for } x,y \in \Bbb C.$$   I have done the proof of the first two propositions for being a metric, but I'm having a problem in proving the triangle inequality. What test can I use to compare slopes from two or more regression models? I would like to test the difference in response of two variables to one predictor. Here is a minimal reproducible example.   library(nlme)  ## gls is used in the application; lm would suffice for this example m.set &lt;- gls(Sepal.Length ~ Petal.Width, data = iris,                 subset = Species == "setosa") m.vir &lt;- gls(Sepal.Length ~ Petal.Width, data = iris,                 subset = Species == "virginica") m.ver &lt;- gls(Sepal.Length ~ Petal.Width, data = iris,                 subset = Species == "versicolor")   I can see that the slope coefficients are different:  m.set$coefficients (Intercept) Petal.Width    4.7771775   0.9301727 m.vir$coefficients (Intercept) Petal.Width    5.2694172   0.6508306  m.ver$coefficients (Intercept) Petal.Width     4.044640    1.426365    I have three questions:   How can I test the difference between slopes? How can I test the difference between residual variances? What is a simple, effective way to present these comparisons?    A related question, , suggests re-running the model with a dummy variable to differentiate the slopes, are there options that would allow the use of independent data sets? Following my question , I am wondering if there are strong views for or against the use of standard deviation to detect outliers (e.g. any datapoint that is more than 2 standard deviation is an outlier).  I know this is dependent on the context of the study, for instance a data point, 48kg, will certainly be an outlier in a study of babies' weight but not in a study of adults' weight.  Outliers are the result of a number of factors such as data entry mistakes. In my case, these processes are robust.   I guess the question I am asking is: Is using standard deviation a sound method for detecting outliers? In , they give rules so that players can make all the rolls, rather than the DM sometimes rolling for enemies. In one section, there are specifically , which the players roll against, rather than the DM. However, after doing some math, I think they've made a mistake. They say that you convert a Saving Throw into a Saving Throw Check by adding 11 to the defender's saving throw modifiers, and using that as the DC for the check.  The player then rolls against this DC, adding their spellcasing ability modifier and proficiency bonus.  However, according to the following math, this conversion does not produce the same results. In this math, I compare the chances of a player succeeding on a saving throw check, using the rules in the UA article, and of a monster failing its saving throw using the standard PHB rules. If the math is correct, both of these results should have the same chance. These are the formulas I used.  Chance to succeed Saving Throw Check = (20 - DC + 1 + proficiency + casting mod) / 20 Chance to FAIL a Saving Throw = (DC - 1 - save proficiency - save mod) / 20  Assuming a proficiency bonus of +2, +0 for all ability mods, and no save proficiency:  Monster's Saving throw Check DC = 11 + save mod = 11 Chance to succeed a saving throw check DC11: (20 - 11 + 1 + proficiency + casting mod) / 20 = 12/20 = 60% Player's Saving throw DC = 8 + proficiency + spellcasting ability mod = 10 Chance for monster to fail saving throw DC10: (10 - 1 - save proficiency - save mod) / 20 = 9/20 = 45%  If this were a valid conversion, both formulas would result in the same chance of success for the player, and failure for the monster respectively, however, they are off by 15%. This suggests that the unearthed Arcana's rules are not a valid conversion. Are the Unearthed Arcana rules wrong, or is my math wrong? Temporarily suspend bash_history on a given shell? Is there a way to temporarily suspend history tracking in bash, so as to enter a sort of "incognito" mode? I'm entering stuff into my terminal that I don't want recorded, sensitive financial info. I found several very-close-to-duplicate questions/answers. However, I want to restate some questions and discuss the answers from these questions.  Here the questions which I found:         I find it quite unfair that super simple questions and their answers upvote give the same amount of reputation points per upvote as complex questions. I think reputation points should be much closer linked to contribution. And I think that contribution in answering/asking a simple question (both in time, value and time saved to other developers) is way less than for some more complex questions.   Why doesn't Stack&nbsp;Overflow change this?  I saw several common answers:     if you change the game, then the only result is that people will game it differently.   I agree only partly with this. Sure, people will always game the system. However, any improvement can be rejected based on this argument. I believe if there is an obvious problem, it makes sense to fix it. And it looks like this question was asked number of times (meaning that it's not only me thinking that it should be changed).     Reputation is NOTHING more than an INCENTIVE to contribute.   Again. I agree only partly. Reputation doesn't have any real value (it can't be converted to anything). However, let's be honest. People are competitive and people like to have some measurement stick and most of software developers want to have a fair game. Based on this, reputation isn't NOTHING, reputation is very important tool which makes Stack&nbsp;Overflow popular. Calculating reputation one way or another way may lure or discourage people from participating.     Esoteric questions are only of interest to people working with that item.   I believe one of the biggest values of Stack&nbsp;Overflow that it contains a lot of information which is hard to find somewhere else.  So, sure, there are some rare questions which aren't valuable to the whole community. But, I would say that a question which saves 25 people a day of work (complex question) is more valuable than a question which saves a 1000 people 3 minutes of googling (simple question).  My summary:  I believe it makes sense to change a system a little bit. The simplest way would be the ability to mark a question as trivial. If some number of people marked the question as trivial (let's say 10), all upvotes for this question/answers become less valuable (for example, 1 or 2 points, instead of 5). This will close this gap. Change elements on different positions within a list What is the integral of 0? Where in the registry does Windows store with which program to open certain file types? After installing Adobe PDF reader, it opens files with the .PDF file extension. Ones' default web browser opens .html files and notepad opens .txt files, etc.  Where exactly in the registry are these filetype - program associations stored in the registry?   In other words, how does Windows know with which software/program it must open a certain file(type)? Can open sets and closed sets of $\mathbb{R}^n$ be homeomorphic? What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? The title pretty much says it all.  Of course, one answer (IMO unsatisfactory) to such questions goes something like "a definition is a definition, period."  In my experience, mathematical definitions are rarely completely arbitrary.  Therefore I figure there must be a good reason for insisting that a field be a non-trivial ring (among other things), but it's not obvious to me.  I realize that a trivial ring would make for a very boring field, but it is also a rather boring ring, and yet it is not disallowed as such.  EDIT: I found a hint of a rationale in the statement that "the zero ring ... does not behave like a finite field," but I could not find in the  for this assertion exactly how the zero ring fails to behave like a finite field. Group with order $p^2$ must be abelian . How to prove that? How to enclosure a text block into big square brackets? everybody.  I need to enclosure a text block into big square brackets.  Something like this (I need to replicate it, more or less):   I've read that it could be possible with tikz but I don't know anything about that package.  Even, I found a code that reproduce something similar but with braces in this post:  The code is this:  \documentclass{article} \usepackage{lipsum} \usepackage[many]{tcolorbox} \usetikzlibrary{decorations.pathreplacing}  \newtcolorbox{rightbrace}{%     enhanced jigsaw,      breakable, % allow page breaks     frame hidden, % hide the default frame     overlay={%         \draw [             fill=none, % fill paper             decoration={brace,amplitude=0.5em},             decorate,             ultra thick,             gray,         ]         % right line         (frame.north east)--(frame.south east);     },     % paragraph skips obeyed within tcolorbox     parbox=false, }  \begin{document}  \begin{itemize}     \item First line           \begin{rightbrace}             \item Second line              \item Third line, which is quite long and seemingly tedious in the extreme             \item Fourth line, which isn't as long as the third            \end{rightbrace}     \item Fifth line \end{itemize}  \begin{rightbrace}     \lipsum[1] \end{rightbrace} \lipsum[2]  \end{document}   How could I put the square brackets around a block of text? (Using tikz, tcolobox or whatever package then can be useful for it).  Thanks for your help. Is it possible to follow a user automatically? We know that the maximum level a dweller can train to on any SPECIAL skill is 10, however this does not consider bonuses given from their outfit.    Is there any additional benefit for going over 10? For example, a dweller with an Intelligence skill of 10 wearing a lab coat that boosts Intelligence by 3?   If it is a hard cap, would it be more useful to use outfits that raise a secondary characteristic instead? Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty$. Prove that $f$ is uniformly continuous.   Since $f(x)\to 0$ as $x\to \pm\infty$ we have $|f(x)|&lt;\epsilon $ whenever $x&lt;-K$ and $x&gt;K$ where $K&gt;0$.  Also $f$ is uniformly continuous on $[-K,K]$ as the domain is compact.  Hence $f$ is uniformly continuous in $(-\infty,-K)\cup [-K,K]\cup (K,\infty)$.  But how to conclude that $f$ is uniformly continuous on $\Bbb R$ from above as I may chose $x\in (-\infty,-K);y\in [-K,K]$ ,how to conclude that $|f(x)-f(y)|&lt;\epsilon $ from above for a given $\epsilon$?  Please suggest the edits required. Theme Image Style for image located in module directory Three side-by-side tables I am using this LaTeX code to generate three tables side by side. However, it showing errors and not giving satisfactory result. Can anyone please help me to short out the problem.  \begin{table} \centering \makebox[0pt][c]{\parbox{1.2\textwidth}{% \begin{minipage}[b]{0.265\hsize}\centering \begin{tabular}{ | c | c |c | } \multicolumn{3}{c} \hline {$q=1/10$} \\ \hline T &amp; $r_h$ &amp; $\omega$ \\  \hline 0.233 &amp; 0.135 &amp; 2.38486-0.34667I  \\  0.256 &amp; 0.140 &amp; 2.38092-0.35837I \\ 0.273 &amp; 0.145 &amp; 2.38049-0.35982I \\  0.278 &amp; 0.146 &amp; 2.38007-0.36130I \\  0.281 &amp; 0.147 &amp; 2.37964-0.36282I \\  \hline 0.284 &amp; 0.761 &amp; 2.54150-2.03209I \\ 0.285 &amp; 0.765 &amp; 2.54515-2.04292I \\ 0.289 &amp; 0.800 &amp; 2.57799-2.13761I \\ 0.302 &amp; 0.900 &amp; 2.67987-2.40712I \\ 0.312 &amp; 1.000 &amp; 2.79214-2.67552I \\ \hline \end{tabular} \end{minipage} \hfill \begin{minipage}[b]{0.265\hsize}\centering \begin{tabular}{ | c | c | c | } \multicolumn{3}{c} \hline {$q=1/9$} \\ \hline T &amp; $r_h$ &amp; $\omega$ \\ \hline 0.227 &amp; 0.150 &amp; 2.34385-0.41094I  \\  0.245 &amp; 0.155 &amp; 2.34215-0.41723I \\ 0.260 &amp; 0.160 &amp; 2.34056-0.42416I \\  0.271 &amp; 0.165 &amp; 2.33908-0.43175I \\  0.276 &amp; 0.167 &amp; 2.33852-0.43497I \\  \hline 0.280 &amp; 0.720 &amp; 2.50248-1.92299I \\ 0.283 &amp; 0.750 &amp; 2.52893-2.00424I \\ 0.286 &amp; 0.780 &amp; 2.55660-2.08533I \\ 0.289 &amp; 0.800 &amp; 2.57568-2.13931I \\ 0.302 &amp; 0.900 &amp; 2.67807-2.40842I \\ \hline \end{tabular} \end{minipage} \hfill \begin{minipage}[b]{0.265\hsize}\centering \begin{tabular}{ | c | c | c |} \multicolumn{3}{c} \hline {$q=1/8$} \\ \hline T &amp; $r_h$ &amp; $\omega$ \\  \hline 0.246 &amp; 0.180 &amp; 2.29733-0.50611I  \\  0.255 &amp; 0.185 &amp; 2.29654-0.51364I \\ 0.263 &amp; 0.190 &amp; 2.29589-0.52172I \\  0.269 &amp; 0.195 &amp; 2.29534-0.53033I \\  0.273 &amp; 0.199 &amp; 2.29496-0.53758I \\  \hline 0.275 &amp; 0.675 &amp; 2.46067-1.80426I \\ 0.277 &amp; 0.700 &amp; 2.48131-1.87194I \\ 0.288 &amp; 0.800 &amp; 2.57245-2.14170I \\ 0.302 &amp; 0.900 &amp; 2.67555-2.41027I \\ 0.317 &amp; 1.000 &amp; 2.78870-2.67796I \\ \hline \end{tabular} \end{minipage}% }} \end{table}