Why is it true that $a^{\log_{b}n} = n^{\log_{b}a}$
I'm trying to understand a "subproof" of the divisor Master Theorem (Cormen et al., Introduction to Algorithms, page 99), and in some point they state:  $$\Large a^{\log_b n} = n^{\log_b a}$$  where $a\geq 1$, $b &gt; 1$ and $n = b^i$, for some $i$. That is, $n \in \{1, b, b^2, ...\}$  I would appreciate a simple explanation of this equality (just OK for a CS undergraduate).
Question is about Ubuntu release process. Are migrations from n to n+1 releases expected?  Assuming we have server with installed Ubuntu 14.04 LTS. Will it be smoothly upgradable to Ubuntu 16.04 LTS via apt-get upgrade ?  Assumptions:   scope is software therefore we assume that given hardware is supported on both releases (upgrading from and to) that's why, always before upgrade you are asked to make backup to be able to revert in case of failure I tried to install Ubuntu 18.10 on my XPS 15 9570 earlier. Everything was working fine until I got to the partition selection part of the installation. That's when the installer crashed and I had to shut down my machine. I think it's because I had my SATA configuration set to RAID ON instead of AHCI, which is now fixed.  Now when I try to run the installer from my bootable USB I get the following error  Failed to open \EFI\BOOT\mmx64.efi - Not Found Failed to load image \EFI\BOOT\mmx64.efi: Not Found Failed to start MokManager: Not Fond Something has gone seriously wrong: import_mok_state() failed   Hoping someone might have some idea as to what is going on Checking to see if a DateTime variable has had a value assigned Real valued analytic function defined on a connected set is constant exponentiation of cardinal number This is a  about DNS (Domain Name Service).   If my understanding of the DNS system is correct, the .com registry holds a table that maps domains (www.example.com) to DNS servers.   What is the advantage? Why not map directly to an IP address? If the only record that needs to change when I am configuring a DNS server to point to a different IP address, is located at the DNS server, why isn't the process instant? If the only reason for the delay are DNS caches, is it possible to bypass them, so I can see what is happening in real time? A Tweedie distribution can model skewed data with a point mass at zero when the parameter $p$ (exponent in the mean-variance relationship) is between 1 and 2.   Similarly a zero-inflated (whether otherwise continuous or discrete) model may have a large number of zeros.  I'm having trouble understanding why it is that when I do prediction or calculate fitted values with these kinds of models, all of the predicted values are non-zero.  Can these models actually predict exact zeros?  For example  library(tweedie) library(statmod) # generate data y &lt;- rtweedie( 100, xi=1.3, mu=1, phi=1)  # xi=p x &lt;- y+rnorm( length(y), 0, 0.2) # estimate p out &lt;- tweedie.profile( y~1, p.vec=seq(1.1, 1.9, length=9)) # fit glm fit &lt;- glm( y ~ x, family=tweedie(var.power=out$p.max, link.power=0)) # predict pred &lt;- predict.glm(fit, newdata=data.frame(x=x), type="response")   pred now does not contain any zeros.  I thought the usefulness of models such as the Tweedie distribution comes from its ability to predict exact zeros and the continuous part.  I know that in my example the variable x is not very predictive. Given $a,b,c$ are the sides of a triangle. Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$ I've had at least one question featured as a "hot network question" and during that time it got thousands of views in the course of a few days. But if you look at that question now you won't have any way of knowing that that's where the bulk of its view count came from.  Is there a way to find out if an arbitrary question has been a "hot network question" or not? How to factor a fourth degree polynomial I'm working on a math problem but I am having a hard time figuring out the method used by my textbook to make this factorization:  $$x^4 + 10x^3 + 39x^2 + 70x + 50 = (x^2 + 4x + 5)(x^2 + 6x + 10)$$  I've tried to see if this equation can be factored by grouping or by long division to no avail. Any help would be greatly appreciated. How to read a text file into GNU R with a multiple-byte separator? What are 'user' and 'system' times measuring in R system.time(exp) output? Can you benefit from the Dueling fighting style for one weapon, then draw a second weapon as part of Extra Attack to qualify for Two-Weapon Fighting? I'm trying to use Linq to return a list of ids given a list of objects where the id is a property. I'd like to be able to do this without looping through each object and pulling out the unique ids that I find.  I have a list of objects of type MyClass and one of the properties of this class is an ID.  public class MyClass {   public int ID { get; set; } }   What I want to do is write a Linq query to return me a list of those Ids  How do I do that given an IList&lt;MyClass&gt; such that it returns an IEnumerable&lt;int&gt; of the ids?  I'm sure it must be possible to do it in one or two lines using Linq rather than looping through each item in the MyClass list and adding the unique values into a list.  Any help in creating an elegant solution would be much appreciated! FTPS versus SFTP versus SCP Regex (C#): Replace \n with \r\n How can I replace lone instances of \n with \r\n (LF alone with CRLF) using a regular expression in C#?  Sorry if it's a stupid question, I'm new to Regex.  I know to do it using plan String.Replace, like:  myStr.Replace("\n", "\r\n"); myStr.Replace("\r\r\n", "\r\n");   However, this is inelegant, and would destroy any "\r+\r\n" already in the text (although they are not likely to exist). How do I install the latest stable version of LibreOffice? LibreOffice is seeing some dramatic improvements each version and subversion that goes out.   I would like to have always the latest version of LibreOffice on my machine(s). I know of the PPA that exists but is it recommended to use it with 12.04 for daily use? What are the pro and cons of using the PPA? Will it break my system/Libreoffice installation?  Does libreoffice get eventually also upgraded in the Ubuntu repositories or does this upgrade happen only with new releases? Who manages the upgrades of libreoffice in Ubuntu and in the PPA? Story where the hero ends up having his eyelids removed and being hooked up to some kind of virtual reality machine Trying to identify a short science fiction story where the protagonist is some kind of policeman or investigator, and at the end he chooses to live his life out hooked up to some kind of virtual reality machine. The ending describes him having his eyelids removed as he is attached to the machine, and then as the machine starts working he "becomes" a cowboy or some other western figure riding his horse downhill in a desert scene. Can I travel to the USA while working remotely for my non-US employer? I would like to travel in the USA for a period of 3 - 6 months. But I want to spend some time working for my employer (remote work) and getting paid by the employer. Is this possible under a tourist visa (B1/B2)?  I hope this is OK since I will not be getting paid by a US employer. I am in the IT trade so all that I will be needing is my own laptop and a connection to the internet. The money I earn would actually help me pay for the expenses I get during the holiday. I'm trying to install queXS cati app on my Ubuntu desktop and I installed MySQL server and PHP 5 and I cannot login into MySQL server as root without password:  mysql -u root    it says  ERROR 1045(28000) : Access denied for user 'root@localhost' (using password: no )   But it's okay when I enter mysql -u root -p   I can't figure out what the problem is.