How can I find ALL complex roots of $z^4 + 1 = 0$?
What are the solutions to $z^4+1=0$?
Sometimes, when I upgrade a system via do-release-upgrade, the update process fails and the script tells me to finish the process manually via dpkg --configure -a. This is a not serious problem - after manually solving the dependencies problem and the finishing upgrade process all works well.  But the do-release-upgrade script has a cleanup process after finishing the upgrade:  Remove obsolete packages?   XXX packages are going to be removed.   Continue [yN]  Details [d]y   which removes some obsolete packages from old version of system.  Seems that this process doesn't execute when the automatic upgrade fails and I finish upgrade manually, so obsolete packages from old version remain installed in the system.  How I can run the "Remove obsolete packages" process manually after finishing the upgrade by hand? How to solve 5x5 grid with 16 diagonals? Blender Internal glossy BSDF equivalent? If every element of set A ia an element of set B then we say that A is subset of B. A is said to be proper subset if A does not equal B.  My question is - whether the symbol we use to show subset is  same as that for proper subset? I have seen that A is subset of A itself is shown by the same symbol as that for the subset while when we write improper subset, we usually write equal to sign with symbol of subset. So, please help me to know this. php - insert a variable in an echo string Is it correct to use "their" instead of "his or her"? Is this sentence grammatically correct?  Anyone who loves the English language should have a copy of this book in their bookcase.  or should it be:  Anyone who loves the English language should have a copy of this book in his or her bookcase. Show that $\frac 1{1+x+y^{-1}}+\frac1{1+y+z^{-1}}+\frac1{1+z+x^{-1}}=1$ if $xyz=1$ If $x.y.z=1$ show that $\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}=1$  My attempt - L.H.S$=\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}$  $=\dfrac y{y+xy+1}+\dfrac z{z+yz+1}+\dfrac x{x+zx+1}$  $=\dfrac{yz}{yz+xyz+z}+\dfrac z{z+yz+1}+\dfrac x{x+zx+1}$  What to do next? Copying prod multi-site installation to dev multi-site installation I am attempting to upgrade my Drupal 8.9.2 multi-site installation to D9. I was hoping to work this all out on a development installation, that is located in a subdirectory of my home directory on my NameCheap Shared Hosting plan. When I first installed this installation, I did so using the tarball method from years ago. I then followed the instructions given here (). I was able to get the current running prod version converted to use composer with out an issue. Before I go and start messing with the files, and attempting to upgrade to D9 through composer on a prod installation, I wanted to copy these files over to another installation where I can use it as a sandbox. To do this I:  copied the files on my server to a new directory made clone databases for the new installation truncated all the cache tables in the cloned DB changed all the database settings in the settings files to point to the cloned databases  I've done this before without any issues, and now I seem to be getting a WSOD error on all the sites running from the new installation. The error I'm getting is  PHP Fatal error:  Uncaught Error: Class 'Drupal\Core\Cache\DatabaseBackend' not found in ~/{new installation dir}/public_html/index.php:16.  If anyone has any thoughts what's going on I am all ears. Thank you in advance Estimator for $\theta$ using the method of moments Exercise :     Using the method of moments, find the estimator for $\theta$ for a random sample $X_1, \dots, X_n$ that follows the distribution with pdf $f(x) = \theta x^{-2}, \; 0 &lt; \theta \leq x &lt; \infty$, where $\theta$ is the unknown parameter.   Attempt :  $$m_1' = \bar{X}$$  $$μ_1' = \int_\theta^xu\theta u^{-2}\mathrm{d}u=\int_\theta^x\theta u^{-1}\mathrm{d}u=\big[\theta \ln u\big]_\theta^x =\theta\ln x - \theta\ln\theta$$  Thus, the estimator is given by :  $$m_1' = μ_1' \Rightarrow \theta\ln x - \theta\ln\theta = \bar{X}$$  But we cannot solve that equation with respect to $\theta$, so we'll take the moments of bigger order :  $$m_2' = \frac{1}{n}\sum_{i=1}^nX_i^2$$ $$μ_2' = \int_\theta^x u^2 \theta u^{-2} \mathrm{d}u = \int_\theta^x \theta \mathrm{d}u = \big[\theta u\big]_\theta^x =\theta x - \theta ^2 $$  Thus the solution of the following quadratic equation for $\theta &gt;0$, gives as the estimator of $\theta$ :  $$\theta^2-x\theta +\frac{1}{n}\sum_{i=1}^nX_i^2$$  Question : Is my solution correct ? Shall the upper bound be $x$ or $\infty$ ? On company 401k matching. Clarification needed I realized I don't fully understand something. It almost embarrassing to admit.  In 2016 we have 18k max contribution limit for 401k. Does this mean that employee can contribute at most 18k or that total contributions made on behalf of employee should not exceed 18k (company match)  To make is simple, if you make 100k, by contributing 18% you got your limit. But if your company matches 5%, actual contributions will be 18.9k. In the above example, if one wants to maximize benefit should one contribute 18% or less? Is $\{\frac{m}{10^n}\mid m,n\in\mathbb Z,\ n\geq 0\}$ dense in $\mathbb R$? The set $S$ of real numbers of the form $m/10^n$, $m,n$ integers and $n$ greater than or equal to $0$, is a dense subset of $\mathbb R$ or not??   I know dense means closure of $S$ in $\mathbb R$ is $\mathbb R$ then it will be dense. How to prove or disprove I have no idea. What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$? text-overflow: ellipsis in table-cell without width "This webpage has a redirect loop" in Chrome but working fine on FF and IE? How can I do case insensitive string comparison in Python?  I would like to encapsulate comparison of a regular strings to a repository string using in a very simple and Pythonic way. I also would like to have ability to look up values in a dict hashed by strings using regular python strings. Okay, this should be super easy to do but I just can't find the way.  I have a background image with a solid colour layer on top of it, and I would like to add text on top of everything in such a way that the filling of the text is the image in the background. I have tried selecting the text shape and deleting that selection shape in the solid colour layer, but it just looks wrong.   Can anyone help me with this? Many thanks in advance. Ordered N-Tuples as Cartesian Products of Indexed Families In Introduction to Topology, Mendelson defines the Cartesian Product of an Indexed Family as follows:     DEFINITION  Let $\{X_\alpha\}_{\alpha \in I}$ be an indexed family of sets. The product  of the sets $\{X_\alpha\}_{\alpha \in I}$,    written  $\prod_{x\in I}X_\alpha$ consists of all functions $x$ with domain the indexing set $I$ having the property that for each $\alpha \in I$, $x(\alpha)\in X_\alpha$.    Halmos, in Naive Set Theory, makes the following observation about this sort of product:     Ordered triples, ordered quadruples, etc., may be defined as families whose index sets are unordered triples, quadruples, etc.   However, it is not immediately apparent to me why this is true. It makes sense if we understand the prior assertions in Halmos that $\prod_{i \in I} X_i = X_i$ and that $\prod_{i \in I} X_i = X_a \times X_b$ when $I = \{a, b\}$, but these already pose difficulty for me.  Let $I = \{0, 1\}$, let $X_0 = \{A, B\}$, and let $X_1 = \{C, D\}$. Then by Mendelson's (and Halmos's) definitions, it would seem to me that . . .  $$\prod_{i \in I} X_ i = \{\{(0, A), (1, C)\}, \{(0, A), (1, D)\}, \{(0, B), (1, C)\}, \{(0, B), (1, D)\}\}$$  This would appear to me to be the set of all functions matching the definition given, but it is not equal to $X_0 \times X_1$. Moreover, extending this to a case where the index set $I$ is a triple, we would not end up with ordered triples, but with each subset of the product containing three ordered pairs.  When Halmos posits the equalities he does, the idea seems easy to grasp, but the peculiar wording of Mendelson's definition suggests something different to me. Making a log4j console appender use different colors for different threads Stack Overflow is not getting refreshed automatically Stack&nbsp;Overflow is not getting refreshed automatically. I have to press F5 explicitly to refresh the website. What might be the issue? How to prove that any (integer)$^{1/n}$ that isn't an integer, is irrational? Is my proof beneath perfect and complete?  I wanted to prove that for any nth root of an integer, if it's not an integer, than it's irrational: $$\begin{cases} m,n\in \mathbb{N}\\\sqrt[n]{m}\notin \mathbb{N} \end{cases}\implies \sqrt[n]{m}\notin \mathbb{Q}.$$  I start by assuming that $m^{\frac 1n}$ is rational and non-integer. So there exist  co-prime integers $a,b$ so that $$\sqrt[n]{m}=\frac{a}{b}$$ $$\implies  m=\frac{a^n}{b^n}\in\mathbb{N}.$$ But since $a$ and $b$ have no common factor, $a^n$ and $b^n$ also have no common factor. So: $$\frac{a^n}{b^n}\notin\mathbb{N},$$ a contradiction.