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Can someone explain this trigonometric limit? I have $$\lim \limits_{x\to 0}  \frac {\tan(2x)}{\sin(x)}$$ and in my case the result is $\frac{2}{1}$ =2 not whether it is right.  This is my procedure.  $$\lim \limits_{x\to 0}  \frac{\frac {\sin(2x)}{\cos(2x)}}{\frac{\sin(x)}{1}}= \lim \limits_{x\to 0}  {\dfrac {\sin(2x)}{(\cos(2x))(\sin(x))}}=\dfrac{2x\frac {\sin(2x)}{2x}}{\cos(2x)\frac{x\sin(x)}{x}}$$  I separate the limit.  $$\frac{\left(\lim \limits_{x\to 0}2x\right) \cdot \left(\lim \limits_{x\to 0}\frac {\sin(2x)}{2x}\right)}{\lim \limits_{x\to 0}\left(\cos(2x)\right)\cdot\left(\lim \limits_{x\to 0}\frac{x\sin(x)}{x}\right)} = \lim \limits_{x\to 0} \dfrac{2x}{x}=\frac{2}{1} =2$$ How do "Dummy Variables" work? I do not understand how dummy variables work in math.   Suppose we have:  $$I_1 = \int_{0}^{\infty} e^{-x^2} dx$$  How is this equivalent to:  $$I_2 = \int_{0}^{\infty} e^{-y^2} dy$$  How does this dummy variable system work?   Since $y$ is the dependent variable for $I_1$ How can $y$ itself be and independent variable for $I_2$  ??  Thanks! show elements of order 15 in $S_8$ satisfy a relation Let $G$ be a group.  Define a relation $\sim$ on $G$ by $a \sim b$ if there exists $g \in G$ such that $a = gbg^{-1}$.  Prove that all elements of order 15 in $S_8$ are related by $\sim$.  I noticed that in order to have order $15$ in $S_8$. We need a $5$ cycle and a $3$ cycle. Then, I am stuck. Can anyone give me some ideas about what to do? Thank you in advance. White screen of death: Fatal error: Allowed memory size of X bytes exhausted I have a issue with my Drupal installation, for example:   when I enable or disable the modules, It redirects me to a blank page, when I create a new content type and save it, redirects me to a blank page, when I add a new view and save it, it redirects me to a blank page, when I try to clear the cache, it redirects me back to a blank page, or in similar cases.   Basically all the confirmation pages redirect me to a white screen. When I refresh it again, it shows me the page. I tried to increase the PHP memory value but it doesn't help.  Are there any other solutions for this?  The error which I'm having:      Fatal error: Allowed memory size of 100663296 bytes exhausted (tried to allocate 8192 bytes) in sites/all/modules/views/plugins/views_plugin_localization_none.inc on line 1 Is centripetal acceleration mandatory for circular motion? How to do research in two very different fields? I’ve always understood that the phrase two weeks usually turns into two weeks’ when used as a modifier as in   I’m giving my two weeks’ notice.  I get two weeks’ vacation. (“two weeks’ holiday” for Brits)   with an apostrophe on the word weeks’, indicating that the vacation “belongs” to the weeks.   One way to explain this is the phrase “two weeks of vacation” being contracted to “two weeks’ vacation” – the vacation is “of” the weeks; that is, it is possessed by them.  But I’ve seen a lot of people omit the apostrophe in casual writing, and thinking about it, it seems plausible that the noun vacation would be plainly modified by the adjectival phrase two weeks.  But on the other hand, shouldn’t adjectival phrases be hypenated, namely two-weeks vacation?  Yet that seems wrong, too: the hyphenate should be singular, two-week vacation (like two-tone shoes) because .   touches on this (and references a good ) but doesn’t explain why the plural must be possessive in this case.   suggests that two weeks might be adverbial rather than adjectival, and thus resists being used attributively. But that still doesn’t quite explain this particular muddle of pluralization, possession, and attribution.  Given all this, why should “two weeks vacation” be written “two weeks’ vacation” with a possessive apostrophe? Seeking methods to solve: $\int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt$ Seeking Methods to solve the following two definite integrals:  \begin{equation} I_(n) = \int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt \qquad J(n) = \int_0^\infty \frac{\ln^2(t)}{t^n + 1}\:dt \end{equation}  For $n \in \mathbb{R},\:n \gt 1$  The method I took was to take the following integral:  \begin{equation}  \int_0^\infty \frac{t^k}{\left(t^n + 1\right)^m}\:dt = \frac{1}{n}B\left(m - \frac{k + 1}{n}, \frac{k + 1}{n} \right) \end{equation}  Where: $0 \leq k \lt n$. Here let $m = 1$.   with respect to $k$ and taking the limit as $k \rightarrow 0^+$ (via the ), i.e.   \begin{align} \lim_{k \rightarrow 0+} \frac{d}{dk} \left[ \int_0^\infty \frac{t^k}{t^n + 1}\:dt \right] &amp;= \lim_{k \rightarrow 0+} \frac{d}{dk} \left[\frac{1}{n}B\left(1 - \frac{k + 1}{n}, \frac{k + 1}{n} \right) \right] \\ \lim_{k \rightarrow 0+}  \int_0^\infty \frac{t^k \ln(t)}{t^n + 1}\:dt  &amp;= \lim_{k \rightarrow 0+}  \left[\frac{1}{n^2} B\left(1 - \frac{k + 1}{n}, \frac{k + 1}{n}\right)\left[\psi^{(0)}\left(\frac{k + 1}{n} \right) - \psi^{(0)}\left(1 - \frac{k + 1}{n} \right)\right] \right] \\ \int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt&amp;= \frac{1}{n^2} B\left(1 - \frac{1}{n}, \frac{1}{n}\right)\left[\psi^{(0)}\left(\frac{1}{n} \right) - \psi^{(0)}\left(1 - \frac{1}{n} \right)\right] \\ &amp;=- \frac{\pi^2}{n^2}\operatorname{cosec}\left(\frac{\pi}{n}\right)\cot\left(\frac{\pi}{n} \right) \end{align}  Which is our expression for $I_n$. Taking the same approach but differentiating twice with respect to $k$ we arrive at our expression for $J_n$:  \begin{equation}  J(n) = \int_0^\infty \frac{\ln^2(t)}{t^n + 1}\:dt = \frac{\pi^3}{n^3}\operatorname{cosec}\left(\frac{\pi}{n} \right)\left[\operatorname{cosec}^2\left(\frac{\pi}{n}\right) + \cot^2\left(\frac{\pi}{n}\right) \right] \end{equation}  And in fact we may generalise:  \begin{equation}     \int_0^\infty \frac{\ln^p(t)}{\left(t^n + 1\right)^m}\:dt = \lim_{k\rightarrow 0}\frac{d^p}{dk^p}\left[\frac{1}{n} B\left(m - \frac{k + 1}{n}, \frac{k + 1}{n}\right)\right] \end{equation}  Where $p \in \mathbb{N}$  This method however was just an extension of another integral. I'm curious, if I had just started with $I_n, J_n$ what alternative methods could be used? Recent movie involving dimensions and time travel/displacement It's a (relatively) recent movie about a kid whose scientist father is missing or presumed dead.  The kid experiences strange situations (e.g., books from a case fall off).  In the end, it was the father the one that was throwing those books to draw the attention of the kid.  Does anyone know the name of this film? Why is the (non-relativistic) stress tensor linear and symmetric? From :     "[...] the stress vector $T$ across a surface will always be a linear function of the surface's normal vector $n$, the unit-length vector that is perpendicular to it. [...] The linear relation between $T$ and $n$ follows from the fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (Cauchy’s equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that the stress tensor is symmetric."   What is the more detailed explanation for this, and/or where can I find it? No CUDA and GPU option I have Blender 2.73a with AMD Radeon 3100 Graphics. I saw  on how to make cycles render faster. It says when I go to User Preferences> System> Compute Device, It has no CUDA option, which means there is no GPU option. How can I fix this? I need to extrude a circular shape from a surface that's basically just a grid of quads. What's the best practice to do this without screwing up the topology too much?  I basically want to have a circular face I can extrude, in order to have a cylinder like protuberance on the object, something like this: Is abstract algebra (mostly?) restricted to $2$-ary operators? This may be due to my own pure ignorance but it's my experience that all abstract algebra I've been introduced to, both in actual courses and in self-studies only exclusively deals with algebraic objects consisting of a set together with one or more binary operators defined on that set, perhaps over some other algebraic structure. Not counting categories here, just "low-level"-ish stuff.  I'm an undergraduate so my knowledge is of course very limited but I can't help to wonder why I never stumble upon algebraic structures with $n$-ary operators? Is there a good reason for this, something along the lines of $n$-ary operators behave "badly" when $n&gt;2$ or is it just my ignorance because I'm a beginner in the field? Transforming the dummy values to be able to take logs I have a panel data model with double-log functional form. I have 4 variables, one of which is a dummy. What is the best way to transform the values of 0 for my dummy to be able to take natural logs (since you can't take logs from zero)? I have found 2 options so far, but I dont know which is the best (or perhaps there is a 3rd option).  1) Transform all the values of 0 into a very small value (0.00000000001) and then take the logs. I'm not sure this wouldn't still change the outcome of my model quite a bit. 2) Change the values of 0 into 1 and the dummy values of 1 into e (the base of natural logarithm). The log of this newly defined dummy then takes on the values of zero and one and the interpretation of my betas would remain the same. How to split the output and store it in an array? This is the output:  3,aac-lc, 93.8, aaclc, 77.3, h.264, 1024.6, h.264, 1029.1, 31, 31, 0,0,0.000000,31,31,0,0,0.000000,7,0,0,0.000000,30,1280 720,10,0,0,0.000000,30,1280 720   I tried with 2 scenario:   Storing in an array    @arr=split(',',$stats);   echo "statistics: $stats"  Storing in a variable   echo $stats | cut -d ',' -f | read s1  echo $s1    But neither scenario is working. How could I select one vertex from my mesh if I know ID of that vertex? I enabled bpy.app.debug = True and I can see all IDs but cant find a specific one. How best to include other scripts? The way you would normally include a script is with "source"  eg:  main.sh:  #!/bin/bash  source incl.sh  echo "The main script"   incl.sh:  echo "The included script"   The output of executing "./main.sh" is:  The included script The main script   ... Now, if you attempt to execute that shell script from another location, it can't find the include unless it's in your path.  What's a good way to ensure that your script can find the include script, especially if for instance, the script needs to be portable? How to correctly install libpng12-0 on the Ubuntu 19.10? For  I need to install libpng12-0 package on the Ubuntu 19.10 system.  I know that it is available .  So I have decided to download this package manually and install it by hand.  On previous Ubuntu versions (such as 18.04 LTS) this trick ends with success, but here on 19.10 it fails in very strange way:   wget http://mirrors.kernel.org/ubuntu/pool/main/libp/libpng/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb sudo apt-get install ./libpng12-0_1.2.54-1ubuntu1.1_amd64.deb  0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B/116 kB of archives. After this operation, 285 kB of additional disk space will be used. Get:1 /home/user/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb libpng12-0 amd64 1.2.54-1ubuntu1.1 [116 kB] Selecting previously unselected package libpng12-0:amd64. (Reading database ... 212864 files and directories currently installed.) Preparing to unpack .../libpng12-0_1.2.54-1ubuntu1.1_amd64.deb ... Unpacking libpng12-0:amd64 (1.2.54-1ubuntu1.1) ...  dpkg: error processing archive /home/user/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb (--unpack):  unable to install new version of '/lib/x86_64-linux-gnu/libpng12.so.0': No such file or directory No apport report written because the error message indicates an issue on the local system                                                                                          Errors were encountered while processing:  /home/user/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb E: Sub-process /usr/bin/dpkg returned an error code (1)   What is wrong here? How can I get libpng12-0 installed on Ubuntu 19.10 system?    Note: the same happens on 19.04, but it is near EOL. Services Actions Missing in Central Admin Why are some muggles allowed to know of the magical world?