I feel like hunt and peck on the Stack Exchange network when choosing the tags for a question. How one should know which tags exist when there is no list or cloud of tags to choose from?
So I really like the tag cloud on Climate Deal. Is there anyway we could get something similar to this on StackOverflow? I don't necessarily mean that we should replace the interesting/ignored tags with this but something like this on the front page would be nice.
Using the properties of real numbers, verify commutativity on complex numbers. I have just started self-studying Linear Algebra after a 3 year hiatus from mathematics (in an extended neuroscience foray), and I want to ensure that I am sufficiently verifying commutativity of complex numbers by using the properties of real numbers.  My attempts...  (1) Commutativity of Addition:   Verify that $w + z = z + w$ for all $w, z \in \mathbb{C}$.  $$w + z = (a + bi) + (c + di) \hspace{10mm}\text{by definition of }\mathbb{C}$$ $$\hspace{13mm} = a + (bi + c) + di \hspace{10mm}\text{by associativity in }\mathbb{R}$$ $$\hspace{17mm} = a + (c + bi) + di \hspace{10mm}\text{by commutativity in }\mathbb{R}$$ $$\hspace{16mm} = (a + c) + (bi + di) \hspace{10mm}\text{by associativity in }\mathbb{R}$$ $$\hspace{19mm} = (c + a) + (di + bi) \hspace{10mm}\text{by commutativity in }\mathbb{R}$$ $$\hspace{13mm} = c + (a + di) + bi \hspace{10mm}\text{by associativity in }\mathbb{R}$$ $$\hspace{17mm} = c + (di + a) + bi \hspace{10mm}\text{by commutativity in }\mathbb{R}$$ $$\hspace{17mm} = (c + di) + (a + bi) \hspace{11mm}\text{by associativity in }\mathbb{R}$$ $$\hspace{13mm} = z + w \hspace{34mm}\text{by definition of }\mathbb{C}$$  $$\blacksquare$$  This seems a bit verbose to me - is it acceptable to combine steps? For example, rather than:  $$w + z = (a + bi) + (c + di) \hspace{10mm}\text{by definition of }\mathbb{C}$$ $$\hspace{13mm} = a + (bi + c) + di \hspace{10mm}\text{by associativity in }\mathbb{R}$$ $$\hspace{17mm} = a + (c + bi) + di \hspace{10mm}\text{by commutativity in }\mathbb{R}$$ can I instead write: $$w + z = (a + bi) + (c + di) \hspace{10mm}\text{by definition of }\mathbb{C}$$ $$\hspace{48mm} = a + (c + bi) + di \hspace{10mm}\text{by associativity and commutativity in }\mathbb{R}$$?  (2) Commutativity of Multiplication:  Verify that $wz = zw$ for all $w, z \in \mathbb{C}$.  $$\hspace{-14mm}wz = (a + bi)(c + di) \hspace{24mm}\text{by definition of }\mathbb{C}$$ $$\hspace{4mm} = a \cdot c + a \cdot di + bi \cdot c + bi \cdot di \hspace{10mm}\text{by distribution in }\mathbb{R}$$ $$\hspace{9mm} = a \cdot c + bi \cdot c + a \cdot di + bi \cdot di \hspace{10mm}\text{by commutativity in }\mathbb{R}$$ $$\hspace{35mm} = c \cdot a + c \cdot bi + di \cdot a + di \cdot bi \hspace{10mm}\text{by multiplicative commutativity in }\mathbb{R}$$ $$\hspace{1mm} = c(a + bi) + di(a + bi) \hspace{20mm}\text{by distribution in }\mathbb{R}$$ $$\hspace{5mm} = (c + di)(a + bi)\hspace{34mm}\text{by distribution in }\mathbb{R}$$ $$\hspace{4mm} = zw \hspace{55mm}\text{by distribution in }\mathbb{R}$$ $$\blacksquare$$  This also seems a bit wonky to me. Any and all suggestions on clearing up my verification, reasoning, etc. would be greatly appreciated! The strange thing about the maximum in Planck's law I read that it makes a difference whether you calculate $\frac{dE(\lambda) }{d \lambda}=0$ or $\frac{dE(\omega)}{d \omega}=0$ in the sense that the maximum energy density with respect to the wavelength does not coincide with the frequency maximum that one would assume to be at $\omega_{max} = \frac{c}{\lambda_{max}}$. Actually, I do not understand why this is so ( Now, I am only interested in a pure physical explanation, mathematically this is clear and due to the chain rule). Does anybody know how to explain this odd thing?  Somehow I feel that the core of the question has not been completely answered. Although I can easily look up wikipedia, where the need for the chain rule is explained, I am rather interested in understanding where the following argument breaks down, which seems to be why this question has caused some confusion in the past(as you can see by googling this question):  So we have $E(\omega)$ the energy radiated at a given frequency. Now this function has a maximum somewhere, so there is a frequency where a maximal amount of energy is emitted. In other words: If you add the sum of the photon's energies at each frequency that are emitted, you will notice that the maximum is reached at this frquency. Now $E(\lambda)$ tells you basically the same for the wavelength, but again: We know where at which frequency the maximal amount of energy is radiated, so we know the corresponding wavelength.   I think there is a need to explain this.   If anything is unclear, please let me know. PhD duration in the USA I am trying to find out what the average duration of PhD studies are in the USA. Here in Spain it used to be 4 years, but it seems that Europe is moving towards a 3-year scheme since students are expected to do their MSc before they start. What is it like in the USA? Is some kind of shortening also occurring? Integrating $\int\frac{x^3}{\sqrt{9-x^2}}dx$ via trig substitution What I have done so far: Substituting $$x=3\sin(t)\Rightarrow dx=3\cos(t)dt$$ converting our integral to  $$I=\int\frac{x^3}{\sqrt{9-x^2}}dx=\int \frac{27\sin^3(t) dt}{3\sqrt{\cos^2(t)}}3\cos(t)dt\\ \Rightarrow \frac{I}{27}=\int \sin^3(t)dt=-\frac{\sin^2(t)\cos(t)}{3}+\frac{2}{3}\int\sin(t)dt\\ \Rightarrow I=18\cos(t)-9\sin^2(t)\cos(t)+c$$ Which is where I'm stuck. I can substitute back in for the $\sin(t)$ terms but don't know hot to deal with the $\cos(t)$ terms and back out information about $x$.   edited: because I miswrote the problem, I am sorry. Ambiguity in Negation: "John did not come because of the rain" How can I test two static methods (one invokes the other) with PowerMockito? I have some plasterboard screws like this:     However, without a drill I have a job getting them into the wall. Am I safe to use normal screws on hangings that aren't going to be taking a great deal of weight? In this case it is a rack of coat hooks. So that I could resize the window to a certain size from within Terminal. How do the radian have a unit? The radian is defined as the ratio of the circumference and the radius. Both are measured in meters. So there should not be a unit for that. But we use 'rad' as the unit of the radian value.   The coefficient of static/kinetic friction also the same, it is a ratio of both forces. Therefore it doesn't have a unit.  So, is there a special reason to have a unit for the radian value? I have gcc 7.3.0 on Ubuntu 18.04. But I am working on a project where I need gcc less than version 6. When I tried to install gcc 5.4 using sudo apt install gcc-5.4, I got this error:  Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package gcc-5.4 E: Couldn't find any package by glob 'gcc-5.4' E: Couldn't find any package by regex 'gcc-5.4'     What is the appropriate way to install gcc 5.4? I am trying to use notepad++ and regular expressions to edit a document. I have text in this format  123.45607894.165@abcd;aba 23.456@av1;084 9871.001@fab:9782581afa   And I need to get only the first part of the line , the one before the @ sign. How would I do that to get this:  123.45607894.165 23.456 9871.001 I have seen people apply a filter to imported data before plotting with pgfplots, see . Is there a way to extend the approach presented there to apply multiple filters? In the MWE below I made the naive modifications to the solution given there, but it seems that only the last filter is applied: the unwanted y=1 data points are plotted too.  \documentclass{article} \usepackage{filecontents} \usepackage{pgfplots}  \begin{filecontents}{foo.txt}   x y z f   0 0 0 0   0 0 1 1   0 1 0 2   0 1 1 3   1 0 0 4   1 0 1 5   1 1 0 6   1 1 1 7   \end{filecontents}  \pgfplotsset{   discard if not/.style 2 args={   x filter/.code={   \edef\tempa{\thisrow{#1}}   \edef\tempb{#2}   \ifx\tempa\tempb   \else     \def\pgfmathresult{nan}   \fi   }}   }  \begin{document} \begin{figure}   \centering   \begin{tikzpicture}   \begin{axis}   \addplot table[x=x, y=f,                  discard if not={y}{0},                  discard if not={z}{0},                 ] {foo.txt};   \end{axis}   \end{tikzpicture}   \caption{$f(x)$ given $y=0$ and $z=0$.}   \end{figure} \end{document} Uniform convergence with two sequences of functions For, $n\ge 1$ let, $g_n(x)=\sin^2(x+1/n)$ , $x\in [0,\infty)$ and $f_n(x)=\int_0^xg_n(t)\,dt.$  Then, which is(/are) correct ?  (A) $\{f_n(x)\} $ converges pointwise to a function $f$ on $[0,\infty)$ but does not converge uniformly on $[0,\infty)$.  (B) $\{f_n(x)\} $ does not converge pointwise to any function on $[0,\infty)$.  (C) $\{f_n(x)\} $ converges uniformly on $[0,1]$.  (D)  $\{f_n(x)\} $ converges uniformly on $[0,\infty)$.  I found that, $$f_n(x)=\frac{1}{2}\left[x-\frac{1}{2}\sin(x+1/n)+\frac{1}{2}\sin(1/n)\right].$$  So, $f(x)=\lim_{n}f_n(x)=\frac{x}{2}-\frac{1}{4}\sin x$.  Now, $|f_n(x)-f(x)|\le 3/4$. So, by M-test, $f_n(x)$ is not uniformly convergent, but point-wise convergent. So option (A) is correct..Am I right?? I'm trying to show that $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$. A hint would be nice. Drawing/Sketching Tool for elaborating questions More often than not I need to elaborate my question put up on Stack Overflow with a custom sketch, UML class diagram, flow chart.    It is in these times I have to open up a custom paint software, factor out some diagrams, export it as an image, attach it to Stack Overflow. I wouldn't know how complicated it is to invest in having such a sketching or drawing tool integrated, It could even be a basic pen stripped off all the advanced features leaving it a bare minimum.  Such a tool if incorporated into the Stack Overflow "ask question" UI would be certainly helpful.  As of Today this Question has been visited around 170 times and 7 answers, This was meant to be the voice of Stack Overflow users. Would you need a Drawing Tool for you on the Stack Overflow site, would it help you to express you questions and answers better? Sadly, only 2, or 3 answers are in the context of the question. The bounty was meant to be a bait and even that ends tomorrow. Hope you'll get the will and the zest to comment on this question today and in the future . Did Dolores Umbridge Have Any Association with Voldemort (or Death Eaters) before His Return? Ejecting a USB 3.0 drive connected to USB 3.0 port reconnects immediately Why is everything floating in space so cold? Of course there is pretty hot stuff too. For example Suns.  But isn't the vacuum of space the perfect containment for heat? And shouldn't the rare collision with particles even heat up an object that floats around in space?  Why aren't they hot from time to time? Is it because they loose their heat by radiant heat?  And if so does radiant heat emission always occur until the object is the same temperature as the surrounding or does it depend on other factors? Finding the left and right cosets of H = {(1), (12), (34), (12) ○ (34)} in S4 I have an exercise where I am supposed to find the left and right cosets of H = {(1), (12), (34), (12) ○ (34)} in S4. But how do I generate the cosets? As I have understood it you are supposed to pick a number that is not in the set H and multiply it with every number in H. But this does not exactly give the right answer. I would really appreciate it if someone gave an easy to understand explanation of how to generate the left and right cosets. launch app, capture stdout and stderr in c++