Is the operator $(\cdot)^{\frac{1}{p}}$ subadditive for $1<p<\infty$ for positive numbers?
Concave implies subadditive
GSC: How to move a site from a domain to sub domain, sub folder or new domain with a sub folder? CodeIgniter Anchor error: URL not found on this server How would I setup  to recognize import arcpy?   I'd like to use PyCharm to develop and debug geoprocessing tool. Linear algebra skew symmetric I can't install or update Apps from Play Store: Error 905. How to fix? Suddenly (after a Play Store update?) I can't install or update any Apps. I always get error code 905 when I try to do so. How can I fix this? An equality involving the mean value property I am trying to establish that  $$\frac2{\pi}\int_0^{2\pi} f(e^{i\theta})\cos^2\left(\frac{\theta}{2}\right)\;\mathrm d\theta=2f(0)+f^\prime(0)$$  for $f$ analytic on $|z| &lt; R,\; R &gt; 1$  I think I am making a mistake somewhere, please point it out to me.Here's what I've done unsing the mean value property and Cauchy's integral formula:  $$2f(0)+f&#39;(0)=2(\frac1{2\pi}\int_0^{2\pi} f(e^{i\theta})d\theta + \frac1{2\pi i}\int_0^{2\pi}\frac{f(e^{i\theta})ie^{i\theta}}{e^{i2\theta}}=\frac{1}{\pi}\int_0^{2\pi} f(e^{i\theta})d\theta\frac1{2\pi}\int_0^{2\pi}\frac{f(e^{i\theta})}{e^{i\theta}}\quad\text{(1)}$$  But when I develop the expression $\frac2{\pi}\int_0^{2\pi} f(e^{i\theta})\cos^2(\frac{\theta}{2}) d\theta$ I get:  $$\frac1{\pi}\int_0^{2\pi} f(e^{i\theta})d\theta+\frac1{2\pi}\int_0^{2\pi}f(e^{i\theta})[{e^{i\theta}}+e^{i-\theta}]d\theta \quad\text{(2)}$$  The second integral in (1) differs from the second integral in (2). Thanks. How do you effectively study physics? How does one read a physics book instead or just staring at it for hours?   (Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.) Why can't pdflatex print EPS figures? How to find a good topic for a PhD research proposal? In many countries an application for a PhD position includes a written research proposal, so my questions is what are some advises/strategies to come up with a good topic/idea for a PhD research proposal and how can one assess the quality/fruitfulness of an idea? As an undergraduate student one just doesn't have the experience to foresee which ideas might have promising research results and which probably won't have. (And I doubt that potential supervisors have the time to comment on every idea of every potential applicant in cases where it is possible to establish some kind of contact before the actual application.) Single quotes vs. double quotes in Python According to the documentation, they're pretty much interchangeable.  Is there a stylistic reason to use one over the other? When is $\phi(n)$ a prime number? I was thinking about this problem today. If $p$ divides $n$, then $p-1$ must divide $\phi(n)$. But $p-1$ is even for all primes greater than $2$, so must $\phi(n) = 2$ - the only even prime? I feel like this argument is shaky and not very rigorous! How can we change root password? I'm going to university in October and thought I'd have a go at a few questions from one of their past papers. I have completed the majority of this question but I'm stuck on the very last part. In honesty I've been working on this paper a while now and I'm a bit tired so I'm probably giving up earlier than I usually would.  I won't write out the full question, only the last part:     Let $$S_r(n) = \sum_{m=0}^n\ (-1)^m m^r {n \choose m}$$ where r is a   non-negative integer . Show that $S_r(n)=0$ for $r&lt;n$. Evaluate   $S_n(n)$.   I have shown that $S_r(n)=0$ for $r&lt;n$ by taking $(1+z)^n= \sum_{m=0}^n\ z^m {n \choose m}$, letting $D_r(f(z))=z\frac d{dz}(z\frac d{dz}...(\frac d{dz}(f(z)))...)$ where $z\frac d{dz}$ is applied $r$ times and applied it to both sides.  The left hand side gives a polynomial, degree n which has factor $(1+z)$ for all $r&lt;n$ and the right hand side gives $\sum_{m=0}^n\ z^m (m)^r {n \choose m}$. Setting $z=-1$ yields the required result.   There is some build up to this, so I'm fairly certain that this was the intended method.  I'm stuck however on the very last part. I have tried finding a form for $D_r((1+z)^n)$, but I'm fairly sure that this isn't the correct approach, as the wording of the question implies that $S_n(n)$ needs to be considered separately.  I'm surprised that I didn't find that this question had already been asked, so apologies if it has been.  Thank you. Richard Stallman states that Ubuntu "has installed surveillance code". See:   Is this true?   I agree with him that "Any excuse Canonical offers is inadequate....". I would have read this about 20-25 years ago.  I believe the subject was an alien individual, but it could have been a non-standard human instead.  The hook of the story was that the subject lacked an organ like the  that allows the halves of the brain to communicate efficiently, allowing the illusion of a single consciousness.  The subject is conscious, and has thoughts, but is subject to hearing commands like "Wash your hands!" "Fetch water!" within its head.  I recall the subject (and others like the subject) believing that the voices are their gods talking to them.  To the reader it becomes clear that this is a separate part of the subject's own mind that is responsible for long-term planning and simply can't communicate with the rest of the mind in the way a "normal" human is accustomed to. Is there a difference between DR/evil and DR/unholy? Store update, insert, or delete statement affected an unexpected number of rows (0) EntityFramework How do I change the extension of multiple files? I would like to change a file extension from *.txt to *.text. I tried using the basename command, but I'm having trouble on changing more than one file.  Here's my code:  files=`ls -1 *.txt`  for x in $files do     mv $x "`basename $files .txt`.text" done   I'm getting this error:  basename: too many arguments Try basename --help' for more information I created a Compute Engine VM (Ubuntu 18.04), and when I log into it using the SSH button in the VM instances list I'm logged in as user username.  After a while, I realised I needed to be able to use another SSH client as well. As per  I enabled OS Login by adding a key-value pair in metadata, where key is enable-oslogin and value is TRUE.  I then tried to add my existing SSH key:  gcloud compute os-login ssh-keys add --key-file path/to/key.pub --ttl 0  which was successful except one major problem: it has added the key for user username_gmail_com and will only allow me to ssh to my VM as that user:  ssh username_gmail_com@myinstance  And it has also created that user on my VM. From the web interface, I'm logged in to the VM with that new user as well (/home/username_gmail_com). Of course, I can still access the original user's data (/home/username). When I disable OS Login the web interface logs me in as username again and SSH from another client is disabled.   Why is the username different when enabling OS Login? Can I change the username with OS Login enabled?   Edit  There has been an addition to the docs explaining exactly this question. Check here:  and on  all the way at the bottom under "Expected login behaviors". How do I free up more space in /boot?