The master method - why can't it solve $T(n) = 2T(\frac{n}{2}) + \frac{n}{\log n}.$ What's the issue with the master theorem? Why is there like different concepts on different channels? What formula applies? what is &quot;polynomial check&quot;? when does the k method apply? There are so many questions in which some say master theorem cannot be applied but some give out the answers. I m pretty confused right now - what to follow?
Apparently recurrences like this cannot be solved with the Master Theorem:  $T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{\log(n)}$  Because $n^{\log_b(a)} = n^1$ is not a polynomial multiple of $f(n) = \frac{n}{\log(n)}$ since $\frac{n}{\frac{n}{\log(n)}} = \log(n)$, which is less than $n^\epsilon$ for any positive $\epsilon$.  But when is this really even an issue?   For example why wouldn't the time complexity be $n \log n$ using case 2 of the Master Theorem? Unrolling the recursion:  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{\log_2(2^k)}$$  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{k}$$  Doesn't this suggest $\Theta(n \log n)$ even though it supposedly cannot be done with Master Theorem?
Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$? We know that $f(x)=x^4+1$ is a .  My question is:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ has a linear factor over $\mathbb{Q}$?   Edit: Thanks for @Jyrki Lahtonen's answer, I want to do some modifications:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?   Thanks in advance! .php files are downloaded instead of being parsed by Apache What is the difference between a class and a set? I know what a set is. I have no idea what a class is.   As best as I can make out, every set is also a class, but a class can be "larger" than any set (a so-called "proper class").  This obviously makes no sense whatsoever, since sets are of unlimited size. It's not like you get up to 25 elements and go "oh, hey, sets are limited to a maximum of 20 elements, so we'll have to call this 25-element thing something else".  Does anybody know what the actual difference between a set and a class is?  Several people have pointed out that you can write paradoxes which involve sets. I get that. But where do classes come into the picture? Finding center and radius of circumcircle Find the center and radius of the circle which circumscribes the triangle with (complex) vertices $a_1,a_2,a_3$. Express the result in symmetric form.   I'm not sure where to start in this question. The circumcenter is the intersection of the perpendicular bisectors, but I don't know how to calculate the line equation of the perpendicular bisectors yet. As for the radius, after we find the circumcenter $c$ we can calculate it using $|c-a_1|$ (or $|c-a_2|$ or $|c-a_3|$; all three should be equal.) So how can I calculate the circumcenter? How are ETF prices calculated after the market is closed For example:  Because the NASDAQ: DAX is tracking the index INDEXDB: DAX should its trading value be completely flat when the market the markets are open in the US, but closed in Germany? I'm not seeing that artifact in any of the charts I can find. Is there a video that explains how Stack Overflow works and can be used? Is there a video that explains how Stack Overflow works and can be used? I.e. a short tutorial to get new users started. If there isn't, then the people behind this site or others who know how it works should make one, because it's overwhelmingly different from other sites and thus difficult for beginners. Translation help review queue? Why does English spelling use silent letters? Questions like  dealing with Ancalagon in particular, and Middle-earth's dragons in general, clearly show that Ancalagon was ridiculously, absurdly, unimaginably enormous.      The non-canonical image above shows one person's impression of how the different dragons of Middle-earth stacked up, with Smaug looking absolutely tiny at the far left, and Ancalagon on the right, too big to even fit in the picture.    Below, a presumably canonical picture, painted by Tolkien himself, shows Smaug in relation to Bilbo.  The rough measurements I took from this image suggest that Smaug is about 11 times longer than Bilbo is tall.  Depending on how tall Bilbo is - almost certainly somewhere between 3'6" and 4' tall - Smaug is approximately 38-44 feet long.      If the first picture is even vaguely accurate, Ancalagon would appear to be a minimum of something like 720 feet long from the tip of his nose to the end of his tail:  the visible portions of his head and body are at least 6 times longer than Smaug is from end to end, and we have to assume that the portion of his body that is out of frame is at least 3 times longer still, perhaps even 4 times or more (so the upper range of his length might exceed 1,000 feet).  Does Tolkien give us any way to estimate how big Ancalagon was?  It isn't his normal practice to describe things in great detail, I realize, but he often gives a few hints that allow for some rough approximation to be made.  Does anything he says suggest that the first image is anything like what he had in mind?  Note:  I haven't read anything about Ancalagon from Tolkien's own pen, except a few quotes on this site;  the most important bit of information I've gleaned is that he smashed 3 volcanoes to pieces when he died (or something like that).  How accurate is this, and what else did Tolkien say?    This question is not a duplicate of "", because none of the answers to that question address Ancalagon's size in any detail - except my own, which is based on what little information I was able to glean from the two pictures I included in this question.  All the other answers address how Ancalagon was killed, not how big he was.  I have no interest in how he died, only how big he was, or rather, what hints we can find that might give some indication of how big he was.  I wouldn't ask this question, linking that question in my own, if it contained the information I am interested in.    Note 2:  Regarding the relative sizes and impressiveness of Smaug and Ancalagon:  Imagine a meeting between Smaug and Ancalagon in Dragon Heaven: Smaug: "How'd you die?" Ancalagon: "A flying boat full of fire made by the Angels and driven by a Magic Elf smashed into me, so I fell onto a mountain and broke it, and I died. The Magic Elf lives in outer space now, and the boat turned into a star. How'd you die?" Smaug: "Um... Some dude shot me with an arrow." I am installing Ubuntu 16.04 in an Intel NUC computer with UEFI enabled. I am using an SSD from another computer. During the installation I chose Erase the disk and make a new full install  At the end of the process I see:  grub-efi-amd64-signed failed installation /target/ Ubuntu 16.04    And the system does not boot.  I tried to use boot-repair but apparently it does not solve the problem. Building the connection between cosine similarity and correlation in R Do we have to tune the number of trees in a random forest? subequations and array in braces I need something like this:  \left\{ first line  (1a) second line (1b) third line  (1c) \right.   all lines should be aligned to left Is there a better alternative to ambiguous for a word with a wide range of meanings, difficult to find, because they become different in connection with person's opinion. No, this is not a , nor is it a solicitation on  to backup or  and only a little bit about Where to backup and When to do it.  This is basically a strategy question: what, where and when all together. "How" is not really relevant: there are tons of back-up programs out there and they all do the same basic thing: make copies of data.  But the real question is: are these back-up programs making copies of the data important to you?  How should you install&amp;run Ubuntu while safeguarding your data?  There is no "one size fits all" when it comes to computers, Ubuntu versions or a backup strategy, therefore I've split this up into 5 basic user types with each their own answer:       (or you're running in UEFI mode!)   And as the above does not fit everyone, one for the impatient:   !    So click one of the above links and go to your user type!  P.S. If you're running a server farm and you've read this question so far, I'm sorry: this is about a single PC, not a network, nor a server farm. Ubuntu 14.04 Wireless not working: no network interface. Atheros [168c:003e] device I just installed Ubuntu 14.04.3 LTS after installing windows 10 in my new VN7-571G and the wireless is not working. It never had: even during the installation it was not recognized. I run a script to gather the data , which is . lspci -knn | grep Net -A2 output 03:00.0 Network controller [0280]: Qualcomm Atheros Device [168c:003e] (rev 20)         Subsystem: Lite-On Communications Inc Device [11ad:0804]  The OS behaves like it has no wifi card. I do not see wifi settings in network settings I'm not a linux power user, so I do not know what to do to solve the issue. What should I do to make it work? I have a CSV file which has postcode data in a field, but it is non-spatial e.g. no Easting and Northing.   I have a shapefile with postcode in a field and some additional data.  I want to join the CSV with the shapefile based on attributes (postcode fields). I know I can do this in ArcMap, but how can I achieve the same thing in QGIS? Access Denied using TakeOwn.exe I have got this file that I can't delete. It happened after a system crash, so the CHKDSK kicked in upon next reboot. After that, I can't delete the file.  THis is on Windows Home Server, and the file is one of those hidden Thumbs.db, and my WHS reports a "File conflict" on the file, the reason: Access Denied"  What I have tried so far, running as an Administrator: Delete: Access Denied TakeOwn.exe : Access Denied Attrib.exe -s -h : Access Denied cacls.exe : Access Denied   (even whe trying to display the current owner) Rename: File not found (its hidden) Re-boot in to safe mode and tried the above: Access Denied Running the above commands as Local System: Access Denied   Tizer Unlocker: Fail (Doesn't find anything locking the file) File Assasin: Crashes ("FindRemoteFileHandles returned NULL value.")  I have used the CHKDSK /f again, rebooted since some suggestions is that the file has been corrupted, but that didn't change anything. Any suggestions ?  Update: In a total separate scenario, I experienced a similar situation; Access Denied when trying to take ownership of a file. It turned out that the file was owned by TrustedInstaler, which is not possible to override just by being an Administrator. In this case, one can use psexec /ids cmd to elevate yourself to System and then TakeOwn.exe /a Prove every odd integer is the difference of two squares I know that I should use the definition of an odd integer ($2k+1$), but that's about it.  Thanks in advance! What developer text editors are available for Ubuntu?