If $a$ and $b$ are roots of the polynom $X^q -  X$, how do you show that $$(a+b)^q = a^q + b^q$$  Thanks!
If $p$ is prime, then $(x+y)^p=x^p+y^p$ holds in any field of characteristic $p$. However all the proofs I have seen use induction and some relatively nasty algebra despite how fundamental this fact seems.  What is the nicest, "highest level proof" you know?
I bought a Minecraft account for my little brother for Christmas, and I'd like to play with him on my server every now and then. If I log in while he's on, however, it kicks him off since we have the same username. Is there any way to change my username so that I can play with him, or anything I can do so that we can both play together using this account on my server?  I'm open to anything that could let us play, whether it be username changes, server property changes, etc. How to parse HTML with PHP? If $|a| = 12, |b| = 22$ and $\langle a \rangle\cap \langle b\rangle \ne e$, prove that $a^6 = b^{11}$ How to calculate the confidence interval of the mean of means? Imagine that you repeat an experiment three times. In each experiment, you collect triplicate measurements. The triplicates tend to be fairly close together, compared to the differences among the three experimental means. Computing the grand mean is pretty easy. But how can one compute a confidence interval for the grand mean?  Sample data:  Experiment 1:   34, 41, 39  Experiment 2:   45, 51, 52  Experiment 3:   29, 31, 35  Assume that the replicate values within an experiment follow a Gaussian distribution, as does the mean values of each experiment. The SD of variation within an experiment is smaller than the SD among experimental means. Assume also that there is no ordering of the three values in each experiment. The left-to-right order of the three values in each row is entirely arbitrary.  The simple approach is to first compute the mean of each experiment: 38.0, 49.3, and 31.7, and then compute the mean, and its 95% confidence interval, of those three values. Using this method, the grand mean is 39.7 with the 95% confidence interval ranging from 17.4 to 61.9.   The problem with that approach is that it totally ignores the variation among triplicates. I wonder if there isn't a good way to account for that variation. Charge controller for Li-ion or Li-Polymer battery I am trying to work out the good charger for Mobile phones, which can serve as good performer for various smart phones out there in market these days &amp; mostly using Li-ion battery. Charging has 4 stages -- "the initial trickle-charging, then constant current charging, then constant voltage charging &amp; finaly Top-up charge" -- all this is controlled &amp; monitored by a control circuit/IC of charger. So, my concern is if such circuit/module/IC is already there in Mobile phones &amp; taking care of all protections to battery/phone including overheating during charge OR The charger designer have to incorporate these circuits in Charger itself? If former case, that means we just need to make a charging source giving ample current &amp; 4.75V-5.25V voltage, like a high-powered USB output port would gives. My feeling is such stages-wise charging &amp; control/monitor circuitry is already within the phone set, i just want to reconfirm. "He didn't know where New Jersey was" Your content must have a ListView whose id attribute is 'android.R.id.list' How to skip kernel update? Sum with parameter convergence The final state of 1000 light bulbs switched on/off by 1000 people passing by There are  1000  light bulbs and 1000 tutors. All light bulbs are off. Tutor 1 goes flipping light bulb 1,2,3,4... tutor 2 then flips 2,4,6,8... tutor 3 then 3,6,9...etc until all 1000 tutors have done this.  What is the status of light bulb 25? 93? 576?  Is there a general solution to find out the status of any light bulb?  How many light bulbs are on after all 1000 tutors have gone by?  Not homework but a UK university interview question. Please help. How to pass Unicode characters as JSP/Servlet request.getParameter? How do I connect to a MySQL Database in Python? How do I connect to a MySQL database using a python program? Do comic characters sometimes get drawn like the actor playing them? I noticed in Spider-Man's recent cameo in Captain Marvel, he appeared an awful lot like Andrew Garfield who is playing Spider-Man in the recent movie.    Irrelevant of whether it is just my imagination, do some depictions of comic characters morph to be more like an actor who plays them around the time a corresponding movie is released?  By comparison, here is Spider-Man's depiction in the Spider-Men crossover: I'm rather interested in getting my feet wet at the interface of  and . Can anybody give me references to some good books on these topics? I'm looking for very introductory technical books. Compute the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$ What would be the sum of the following series?  $$\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$$  Thanks Java: Date from unix timestamp This error occurs whenever I try to work with sudo. I changed the permissions of /var/ folder to 777 and I think this causes the error. Please help me get back a functional sudo and /var. Using notify-send with cron Schengen visa refusal on threat to public policy, security, health I applied for German Schengen visa as a medical escort to my relative in a different country from  where I reside, but my visa application was refused and the refusal letter  was written in German as the following:        Ein oder mehrere Mitgliedstaaten sind der Auffassung, dass Sie eine Gefahr für die öffentliche Ordnung, die innere Sicherheit, die   öffentliche Gesundheit gemäß Artikel 2 Absatz 19 der Verordnung (EG)   Nr 562/2006 (Schengener Grenzkodex) oder die internationalen   Beziehungen eines oder mehrerer Mitgliedstaaten darstellen.      I was in Germany for nearly a year with my relative during his treatment; I extended my stay there 3 times; and I returned back home two months ago in order to renew my passport. I applied again using the same documents that I used for my extension in Germany.    First, can anyone tell me what the refusal means? Next, is there any possibility that there is a mistake in the assessment?    Note: In English this reason is...     One or more member state(s) consider you to be a threat to public   policy, internal security, public health as defined in Article 2(19)   of Regulation (EC) No 562/2006 (Schengen Borders Code) or the   international relations of one or more of the member states). Describe the set of points on the complex plane...