How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
What is necessary and/or sufficient for polynomials to provide isomorphic quotient rings? Proving $\int_{0}^1\frac{x-1}{\log x}dx=\log 2$ and $\int_{0}^1\frac{\log x}{x-1}dx=\frac{\pi^2}{6}$ I would like to know how to prove the following two definite integrals.  A: $$\int_{0}^1\frac{x-1}{\log x}dx=\log 2$$  B:$$\int_{0}^1\frac{\log x}{x-1}dx=\frac{\pi^2}{6}$$  I found these two by using wolfram-alpha, but I can't prove them. I suspect that the following relations might be used.  $$\log 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}$$and$$\frac{\pi^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots=\sum_{n=1}^\infty\frac{1}{n^2}$$  I need your help. 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? Regaining control of a group after being removed by another admin I created a Facebook group and was later removed by another admin.   How do I get my group back? Per-site Meta post didn't populate Moderator Inbox Per  I'm posting this issue.  I'm a moderator over on Sci-fi &amp; Fantasy.SE.  My moderator inbox missed :    The Meta Question in question should show up after the moderator message and before the new meta question, as it was posted on Jan 2 (it's currently Jan 6, so '2 days ago' is Jan 4). How to make child process die after parent exits? Suppose I have a process which spawns exactly one child process. Now when the parent process exits for whatever reason (normally or abnormally, by kill, ^C, assert failure or anything else) I want the child process to die. How to do that correctly?    Some similar question on stackoverflow:   (asked earlier)  (asked later)      Some similar question on stackoverflow for Windows: Does anybody know of a method to trigger an event in Prototype, as you can with jQuery's trigger function?  I have bound an event listener using the observe method, but I would also like to be able to fire the event programatically.  Thanks in advance We are always introduced to magnets by pictures illustrating a parallepiped magnet, a horseshoe magnet, etc... Anyway the first time we actually calculate a magnetic field is for eelectromagnets. Indeed I've rarely seen explaining or calculating the magnetic field of a permanent magnet. I know formulas for the magnetic field of a moving charge, around a conductive wire, inside a solenoid, but I don't have idea on how to go about calcualting the filed of a simple permanent magnet. It's like that without the help of electricity I'm stuck. I'd guess it would be heavily dependent on the magnetic material, its shape and the medium in which is placed. I hope that someone would provide me some insights on why such formulas are  never introduced in most physics courses and lay down some examples of a permanent magnet field caluclation. How to show $d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}$ is a metric? $d_1(x_1,y_1)$ and  $d_2(x_2,y_2)$ are metric on $X$ and $d(x,y)$ is defined as: $$d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}.$$ I am trying to show this is a metric. Can you give me some clue about proving the triangle inequality for $d$ ? Thank you for your help. Reduction Integration of $I_n=\int{\frac{x^n}{\sqrt{x^2+a^2}} dx}$ How do I derive this using reduction formula?  $$I_n=\int{\frac{x^n}{\sqrt{x^2+a^2}} dx}$$  I got $\sqrt{x^2+a^2}(x^n) - \frac{1}{n-1}(x^{n-1})\sqrt{x^2+a^2} + \frac{1}{n-1}(I_n) $ . I'm not too sure if this is correct. I would appreciate any form of help or correction to my answer. Thank you. Anyone know a quick easy way to migrate a SQLite3 database to MySQL? Weapon damage bonuses and critical hits Why is "be" here? People be like I'm on Ubuntu 16.04.1 at the moment, and system windows like the Files window, Terminal or System Software open behind the Unity launcher, which means they are partially hidden behind it. I can move them with Alt+F7, but I'd like a more permanent solution to this problem. I'm considering replacing an outdoor lighting fixture with an LED motion-sensitive fixture, but I'm not sure how to safely do the wiring.  The location is protected (under a porch) but still needs to be weatherproof. Here's the existing fixture and mount:    The obvious replacement would be another lamp that included its own box cover, so that its wiring would be covered and protected. But, the fixture I like doesn't include such a cover:    You can see it's meant to be screwed into the wall, but doesn't cover anything. The power cable ends with bare wires which are meant to be wire-nutted to the supply wiring, probably inside a junction box. But, how do I get the wire safely into the junction box while keeping things weatherproof?  One option would be to replace the lamp's power cable with one ending in a grounded plug, and install a weatherproof outlet box. But that's ugly and complex.  Are there weatherproof junction box covers that include provisions for sealed entry of cables? How to find number of prime numbers between two integers I have two integers, x and y so that x &lt; y. How many prime numbers are there between x and y (exclusive). Is there a formula or algorithm to compute? How are constraint forces represented in Lagrangian mechanics? In how many ways can we choose k items from n distinct items put in a circle.   Then further extension of the question, if I want to find out the number of ways to choose k items from n distinct items in a circle such that no 2 are together.  Thanks IIS7 - The request filtering module is configured to deny a request that exceeds the request content length Intuitive Explanation of Tippe Top Effect?