radius of convergence of number of divisors
Radius of Convergence of a given Power series
I have two Python dictionaries, and I want to write a single expression that returns these two dictionaries, merged (i.e. taking the union).  The update() method would be what I need, if it returned its result instead of modifying a dictionary in-place. &gt;&gt;&gt; x = {'a': 1, 'b': 2} &gt;&gt;&gt; y = {'b': 10, 'c': 11} &gt;&gt;&gt; z = x.update(y) &gt;&gt;&gt; print(z) None &gt;&gt;&gt; x {'a': 1, 'b': 10, 'c': 11}  How can I get that final merged dictionary in z, not x? (To be extra-clear, the last-one-wins conflict-handling of dict.update() is what I'm looking for as well.) Testing for linear dependence among the columns of a matrix I have a correlation matrix of security returns whose determinant is zero. (This is a bit surprising since the sample correlation matrix and the corresponding covariance matrix should theoretically be positive definite.)  My hypothesis is that at least one security is linearly dependent on other securities. Is there a function in R that sequentially tests each column a matrix for linear dependence?  For example, one approach would be to build up a correlation matrix one security at a time and calculate the determinant at each step. When the determinant = 0 then stop as you have identified the security who is a linear combination of other securities.  Any other techniques to identify linear dependence in such a matrix are appreciated. Is there a faster way to open a process as Administrator (With UAC enabled)? Currently (In Windows 10) - if I want to open a CMD prompt, I'll hit the Windows key, type CMD and hit Enter. If I want it open as an administrator, I have to right click the item and click Run As administrator. Is there a way I can do this without using the mouse? What is the solution to the following recurrence relation with square root: $T(n)=T (\sqrt{n}) + 1$? Why is Apache executing .php.html files as PHP? I have an odd problem...I'm using a documentation generator which generates a lot of output like docs/foo.php.html. It's XHTML, and thus contains &lt;?xml...&gt; tags at the beginning of file. The problem is, Apache has somehow decided to run it through the PHP interpreter, even though ".php" appears in the middle of the filename, and not at the end. This, in turn, triggers a PHP error, because it sees "&lt;?" as the command to start executing PHP code, and immediately gets confused by the "xml..." which follows it.  How do I configure Apache to ONLY execute .php files and not .php.html files? The string "php.html" does not appear explicitly anywhere in my Apache config files. There is a line "AddHandler php5-script .php", but I don't see how that would also include ".php.html" files. t-test for partially paired and partially unpaired data An investigator wishes to produce a combined analysis of several datasets.  In some datasets there are paired observations for treatment A and B.  In others there are unpaired A and/or B data.  I am looking for a reference for an adaptation of the t-test, or for a likelihood ratio test, for such partially paired data.  I am willing to (for now) to assume normality with equal variance and that the population means for A are the same for each study (and likewise for B). Give moderators the ability to mark reviews as "bad" From what I've seen, many SO moderators trawl completed reviews and watch out for robo reviewers; and manual banning them when they see enough (>= 3?) bad reviews. This is inefficient because:   It doesn't integrate into the audit system. There's no "memory" to it. If mod A finds that user X has made two (any number less than the threshhold) bad reviews, he may make a mental note but there probably won't be any permanent/shared memory of this. If moderator B finds this user at a later date, he will have to look through the entire history, including stuff that mod A has already checked.    What I propose is that moderators get the ability to mark individual reviews as "invalid" from the completed review page. This is only to be used for reviews where any reviewer who is paying attention (and knows the rules) would choose the opposite reason (spam and other low quality stuff).   These bad review flags are treated like audits for a user -- if the number of failed audits and bad reviews is enough, the user gets autobanned. The nice thing is that this system should (probably) easily be able to , fixing the "Why was I manual banned?" problem.  This could also be integrated well with . Trying to identify pre-1970s short story involving multiple humans absorbed into an alien blob Where is the "Manipulate Centers Only" button in Blender 2.8? I have looked just about everywhere on the Internet and I can't find the answer to this question. Where the hell is the "Manipulate Centers Only" button in Blender 2.8???? It's the the double arrow icon here:    I seem to have enabled it somehow and I can't scale or rotate my object. Searching for words in question titles I'm trying to search for questions that have specific words in their titles using the  search feature, but it seems that it does not return any search results.  Using this feature, I tried to search for questions on Stack Overflow with , but this search query returns no results.  Is it still possible to filter the search results based on words in question titles? e.g., I've got files I want to send by sendmail but the receiving mailbox (for posting pictures to my blog) will not accept uuencode files a la   $uuencode file | sendmail user@domain  I'm guessing it'll accept MIME though. So what command can I use instead? $\gcd(b^x - 1, b^y - 1, b^ z- 1,\dots) = b^{\gcd(x, y, z,\dots)} -1$ Possible Duplicate:        $b$, $x$, $y$, $z$, $\ldots$ are integers greater than 1. How can we prove that  $$ \gcd (b ^ x - 1, b ^ y - 1, b ^ z - 1 ,\dots)= b ^ {\gcd (x, y, z, \dots)} - 1\ ? $$ How to calculate crc16 in php Proof Check: Every Cauchy Sequence is Bounded Sorry if I keep asking for proof checks. I'll try to keep it to a minimum after this. I know this has a well-known proof. I understand that proof as well but I thought I'd do a proof that made sense to me and seemed, in some ways, simpler. Trouble is I'm not sure if it's totally correct. It's quite short though. I was just hoping someone could look it over and see if it is a valid proof. Thank you!  Lemma: Every Cauchy sequence is bounded.  Proof: Let $(a_{n})$ be Cauchy. We choose $ 0&lt;\epsilon_{0}$. So $ \forall \; n&gt;m\geq N_{0}$ we have that $\vert a_{n}-a_{m} \vert &lt; \epsilon_{0}$. Therefore $(a_{n})$ is bounded for all $ m \geq N_{0} $ by $ \epsilon_{0} $. Since $ \mathbb{N}_{N_{0}}$ is finite, it is bounded. So, for all $ m&lt;N_{0} $, $ (a_{n})$ is bounded. Therefore $(a_{n})$ is bounded.  I realize I haven't said what the bounds are but I think that's sort of irrelevant. So long as we know it's bounded. Any help is much appreciated! Is it possible to know from apt if the package was installed explicitly or as a dependency of another package? A real nxn matrix always has a real eigenvalue when n is odd what do it mean by "A real nxn matrix always has a real eigenvalue when n is odd?" Shouldnt it be that regardless of odd or even there will always have a real root?  hope to hear from you guys soon:) I was reading an example about describing homomorphisms and I'm having a bit of trouble with this one.  Show there is no group homomorphism $f : \mathbb{Z}_{10} \to \mathbb{Z}_{25}$ such that $f(1) = 3$. The first step of the proof is  $|1| = 10$ in $\mathbb{Z}_{10}$, so $|f(1)| \in\{ 1, 2, 5, 10\}$.  Does this mean that $|f(1)|$ can be $1$, $2$, $5$ or $10$? I see that $|1| = 10$ since $1$ is a generator of $\mathbb{Z}_{10}$, and that $|f(1)|$ seems to be a factor of $10$. If $f(1)$ is the image of $1$ under $f$, how would I find the order of $f(1)$?  Thanks for your help. How to activate hyphenation in \ttfamily without explictly setting a font? I am suprised that in the code below no hyphenation is done. And I can not find something about that on the web. Maybe I used the wrong words for searching.  \documentclass{scrartcl} \usepackage{xltxtra} \usepackage{polyglossia} \setdefaultlanguage[spelling=new]{german} \usepackage{tabu}  \begin{document} \begin{ttfamily} \begin{tabu} spread \textwidth {X[l]|X[l]|X[l]|X[l]}     A&amp;     B&amp;     Sichtweise, Haltung, Akzeptanz, useability, Anwenderfreundlichkeit&amp;     D\\ \end{tabu} \end{ttfamily} \end{document} The startup disk cannot be partitioned or restored to a single partition. Windows 10 and Mojave The startup disk cannot be partitioned or restored to a single   partition.      The startup disk must be formatted as a single Mac OS Extended   (Journaled) volume or already partitioned by Boot Camp Assistant for   installing Windows.   I had installed Windows 10 via bootcamp in my mac. Current version is 10.14.1  Now I don't want windows and need that space back but getting above error when I go bootcamp to remove it.  This is the output of diskutil list  /dev/disk0 (internal, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *251.0 GB   disk0    1:                        EFI EFI                     209.7 MB   disk0s1    2:                 Apple_APFS Container disk1         181.3 GB   disk0s2    3:       Microsoft Basic Data BOOTCAMP                60.3 GB    disk0s3    4:           Windows Recovery                         509.6 MB   disk0s4  /dev/disk1 (synthesized):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      APFS Container Scheme -                      +181.3 GB   disk1                                  Physical Store disk0s2    1:                APFS Volume Macintosh HD            116.4 GB   disk1s1    2:                APFS Volume Preboot                 44.6 MB    disk1s2    3:                APFS Volume Recovery                512.8 MB   disk1s3    4:                APFS Volume VM                      4.3 GB     disk1s4   And this is of diskutil cs list  No CoreStorage logical volume groups found   Can anybody please help me to solve this problem?  Output after I tried commands given in the first answer  ❯ diskutil list /dev/disk0 (internal, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *251.0 GB   disk0    1:                        EFI EFI                     209.7 MB   disk0s1    2:                 Apple_APFS Container disk1         250.8 GB   disk0s2  /dev/disk1 (synthesized):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      APFS Container Scheme -                      +250.8 GB   disk1                                  Physical Store disk0s2    1:                APFS Volume Macintosh HD            116.4 GB   disk1s1    2:                APFS Volume Preboot                 44.6 MB    disk1s2    3:                APFS Volume Recovery                512.8 MB   disk1s3    4:                APFS Volume VM                      3.2 GB     disk1s4 Why do we say that the Earth moves around the Sun?