Paladin/Warlock sharing spell slots
What level are spells cast at if a warlock multiclasses as another caster?
Boy and girl paradox is driving me crazy $\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \mathrm {d}x$ Evaluate Integral Thor: Ragnarok ends with the remaining people of Asgard aboard a refugee ship that meets Thanos' ship. Avengers: Infinity War opens with      Thanos and his minions having commandeered the ship and several bodies can be seen on the floor.   This begs the question, what happened to these characters:   Korg Miek Asguardian extra #3 Valkyrie Theorem 13.9 in these  is the polynomial spectral mapping theorem:     Theorem 13.9 For a polynomial $p$ we have $\sigma(p(T)) = p(\sigma(T))$.     My questions:   First direction: Why would $q(T) (p(T) - p(\lambda) I)$ be an inverse for $T - \lambda I$? Update: @RobertIsrael said that instead $(p(T) - p(\lambda) I)^{-1} q(T)$ should be $(p(T) - p(\lambda) I)^{-1}$ be? According to the lines in the proof above, we have $$ (p(T) - p(\lambda) I)^{-1} = ((T - \lambda I) q(T))^{-1} \overset{?!}{=} q(T)^{-1} (T - \lambda I)^{-1}, $$ but as $\lambda \in \sigma(T)$ by assumption, $(T - \lambda I)^{-1}$ doesn't exist, right? For the second direction: Why does $x_i \in \sigma(T)$ imply that $p(x_i) - \mu = 0$? I thought that $x_i \in \sigma(T)$ only implies that $T - x_i I$ is not boundedly invertible. Since $\mu \in \sigma(p(T))$ we have that $p(T) - \mu I = c \prod_{k = 1}^{n} (T - x_i I)$ is not boundedly invertible but how does that help?   In both statements there seem to be a notion of "a product of operators is not invertible iff all factors aren't invertible", but how can that be true? Consider i.e. shift operators on a sequence spaces such that $ST = I$, i.e. the left and right shift operators, then the above mentioned notion doesn't hold, right? How to list files in a zip without extra information in command line In my bash command line, when I use unzip -l test.zip I get the output like this:  Archive:  test.zip   Length      Date    Time    Name ---------  ---------- -----   ----    810000  05-07-2014 15:09   file1.txt    810000  05-07-2014 15:09   file2.txt    810000  05-07-2014 15:09   file3.txt ---------                     -------   2430000                     3 files   But I am interested only by the lines containing the file details.  I tried to make filtering using grep like this:   unzip -l test.zip | grep -v Length | grep -v "\-\-\-\-" | g -v Archive | grep -v " files"  But it is long and prone to error (e.g a file name Archive in this list will be dropped)  Is there any other options with unzip -l (I checked the unzip man page and did not find any) or another tool to do so?  It is important to me to not really unzip the archive but just to look what files are inside. Correct the "about" page to reference the new "on hold" policy The first given name of an author or editor in my bibliography should be written out, but all further first names should be abbreviated (initials with a period -- and space if there are multiple first names/initials, see below). I am using style=authoryear with BibLaTeX (v. 2.7) and Biber (v. 1.7).  My bibliography looks like this:     Doe, John Arthur Kyle (2013). The secret life of John Doe.   It should look like this:     Doe, John A. K. (2013). The secret life of John Doe.   Minimal example:  example.tex  \documentclass{scrartcl}  \usepackage[ngerman]{babel} \usepackage[babel,german=quotes]{csquotes} \usepackage[backend=biber,style=authoryear]{biblatex} \bibliography{bibliography}  \begin{document}  \cite[123]{Doe} \printbibliography  \end{document}   bibliography.bib  @book{Doe,     title = {The secret life of John Doe},     author = {Doe, John Arthur Kyle},     year = {2013} }   (Worked with said versions of BibLaTeX and Biber, Texmaker (v. 4.0.3) and MiKTeX (v. 2.9.4813).)  I could not find a solution in biblatex.def and firstinits=true abbreviates all first names, including the first one. A variation on the induced topology (subspace topology) How to list files installed by a snap package? After a snap has been installed with sudo snap install [package] how can I list the files which have been installed by the snap? Will higher temperatures damage a computer over time, even when not reaching the maximum temperature? For example, if the maximum is 105 degrees celcius, will the computer break faster when continually having it on 80 degrees celsius instead of, for example, 60 degrees celcius. I got a paged loop that pulls in 4 posts per page from custom post type. It works great, with the only downside that the pagination wont go over the page/20 everything after that is 404, and the posts_nav_link won't show the next arrow at all. Like the pages just aren't there.  In total there are 119 published posts in the custom post type. If I change posts_per_page to, say 6, then I get access to more of the posts from the end, but it also stops at the page/20.  On the production site the pagination stops at 21, I guess because the live database has more entries.  I got the "Blog pages show at most" setting set to 6.  Here's the loop I'm using:  &lt;?php $paged = get_query_var('paged') ? get_query_var('paged') : 1; $args  = array(     'posts_per_page' =&gt; 4,     'orderby'        =&gt; 'modified',     'post_type'      =&gt; "projekte",     'paged'          =&gt; $paged ); $custom_query = new WP_Query($args); $c = 0; while($custom_query-&gt;have_posts()) :     $custom_query-&gt;the_post();     ?&gt;     &lt;article&gt;...&lt;/article&gt;     &lt;?php endwhile; wp_reset_postdata(); // reset the query ?&gt;   I've gone through the functions.php a million times looking for a some limiting get_pre_posts filter but there's nothing I can see.  I also got an error log setup and nothing is returned there, I've tested it that it works.  Any suggestions? How can I gzip standard in to a file and also print standard in to standard out? I want to execute a command, have the output of that command get gzip'd on the fly, and also echo/tee out the output of that command. i.e., something like: echo &quot;hey hey, we're the monkees&quot; | gzip --stdout &gt; my_log.gz  Except when the line executes, I want to see this on standard out: hey hey, we're the monkees I can run faster than _____. (1) him (2) he? Pass extra parameters to an event handler? Predict probabilities from Firth logistic regression in R I'm using the logistf package in R to perform Firth logistic regression on an unbalanced dataset. I have a logistf object:  fit = logistf(a~b)   Is there a predict() function like on that's used in the lm class to predict probabilities for future data points? Or do I have to manually input the estimated parameters from the Firth regression. Assume I have a dataset for a supervised statistical classification task, e.g., via a Bayes' classifier. This dataset consists of 20 features and I want to boil it down to 2 features via dimensionality reduction techniques such as Principal Component Analysis (PCA) and/or Linear Discriminant Analysis (LDA).  Both techniques are projecting the data onto a smaller feature subspace: with PCA, I would find the directions (components) that maximize the variance in the dataset (without considering the class labels), and with LDA I would have the components that maximize the between-class separation.  Now, I am wondering if, how, and why these techniques can be combined and if it makes sense.  For example:   transforming the dataset via PCA and projecting it onto a new 2D subspace transforming (the already PCA-transformed) dataset via LDA for max. in-class separation   or    skipping the PCA step and using the top 2 components from a LDA.   or any other combination that makes sense. Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$  Now, here I got stuck. How to prove it?? Please help. When I visit the reputation tab of my profile, I see a lot of stuff. I can make out some of it, but not all.   What do all the event types mean? Why do some events show an odd reputation gain or not show any reputation change at all? Why are some events sometimes highlighted in yellow? My reputation suddenly dropped, but I don't see anything. What happened? Why do some events disappear from my reputation history? Can someone tell me what's wrong with this? because i can't figure out where the missing \right( is.  \begin{equation}  \begin{split} Μ=&amp;-\pi \rho \frac{c^{2}}{4}    \left[c\left(\frac{α}{c}-\frac{1}{2}\right)\ddot{h}+Uc\left(\frac{3}{4}-     \frac{α}{c}\right)\dot{\alpha} +\frac{c^{2}}{4}\left(\frac{9}{8}+      \frac{4α^{2}}{c^{2}}-\frac{4α}{c}\right)\ddot{\alpha} \\  &amp;-\left(\frac{4α}{c}-1\right)U C(k) \left(-\dot{h} +U\alpha + c \left(\frac{3}{4}-\frac{α}{c}\right)\dot{\alpha}\right)\right]  \end{split} \end{equation} Conditions for vertical circular motion If a bob attached to a string is swinging in vertical circular motion, we say that the bob leaves the circular path when the tension in the string becomes zero, but even when tension becomes zero, there's always a component of the bob's weight towards the center of the circle. Why then does the bob leave the circular trajectory?