Matrix library for large matrices?
What are the most widely used C++ vector/matrix math/linear algebra libraries, and their cost and benefit tradeoffs?
Group of positive rationals under multiplication not isomorphic to group of rationals Showing that $e^z=z$ has infinitely many solutions One of our review problems for my complex analysis final is the following:     Show that $e^z=z$ has infinitely many solutions in $\mathbb{C}$.   I've seen a few solutions online, but none of them have used material that was covered in our introductory complex analysis course. Would anyone be able to provide a simple solution?  I'm trying to think of ways to invoke bounding in some way but I can't seem to come up with anything. How do I cherry pick packages from a PPA? What motivates people to answer questions in Stack Overflow? I want that too - motivation to answer programming-related questions. What is yours? Browsing through a copy of Counterexamples in Analysis (Gelbaum  &amp; Olmsted), I'm quite enjoying the typeface used for 'script' letters (see links to examples below); visually it's somewhere between what I'd now call calligraphic and fraktur, and semantically it's used where now we'd use mathcal, mathfrak or mathbb.  I was therefore procrastinatiously looking for a LaTeX-usable font to include these characters, but couldn't seem to find any. I'm certain they're not unique to this text or publisher, so figured my lack of success was probably more to do with searching with the wrong keywords. If anyone has any ideas I'd be grateful. CodeIgniter/PHP Active Record won't increment an integer What is the definition of a symmetric distribution? What's the definition of a symmetric distribution? Someone told me that a random variable $X$ came from a symmetric distribution if and only if $X$ and $-X$ has the same distribution. But I think this definition is partly true. Because I can present a counterexample $X\sim N(\mu,\sigma^{2})$ and $\mu\neq0$.  Obviously, it has a symmetric distribution, but $X$ and $-X$ have different distribution! Am I right? Do you guys ever think about this question? What's the exact definition of symmetric distribution? Is there any lingual way to differentiate between the family that I’m a child of and the family that I am a parent of? (I.e., the first family consists of my parents, my siblings, and me; the second one consists of my spouse, my children and me.) Run a GUI program at startup I am using Ubuntu 14.04. I wrote a small script named trial. The contents of the script are as follows:   #!/bin/sh SHELL=/bin/sh PATH=/bin:/sbin:/usr/bin:/usr/sbin      sh -c firefox   I copied the script to /etc/init.d, modified permission using chmod +x trial and used update-rc.d trial defaults. The file made link but when I rebooted the machine. It did not run firefox. I tried cron @reboot but with no success. I tried rc.local too again failure. When, if ever, should voting override answer acceptance for answer sorting? How to check type of variable in Java? Does each new Horcrux take half of Voldemort's remaining soul, thus taking a smaller piece each time? When you commit a murder and create a Horcrux, you split your soul. It’s been talked a lot about Voldemort splitting his soul in 7/8 parts. But I think his remaining soul was not 1/7 or 1/8 but far more weaker than that because he did not create all Horcruxes at once.  When he created his first Horcrux, his soul was split into 50-50 (diary). But when the second Horcrux was created, it was created by Voldemort only and the first Horcrux soul should have remained intact. Hence it would have been a 25-25 split of Voldemort's main soul. And so on for every split.  Am I right? Sort an array without rewriting a key How do I test a private function or a class that has private methods, fields or inner classes? How do I unit test (using xUnit) a class that has internal private methods, fields or nested classes? Or a function that is made private by having  (static in C/C++) or is in a private () namespace?  It seems bad to change the access modifier for a method or function just to be able to run a test. Why my DCIM folder is empty seen in the computer but not empty in the phone I am using HuaWeiNova device, and I tried to access it with aft-mtp-cli. $ aft-mtp-cli $ ls DCIM/  I got folders listed here: |-photos |-Camera ....  When I tried to list or download my photos in the Camera/ folder, I got nothing. Only the empty folders are listed and the folders that contains files are not shown at all. I am quite sure my photos are stored in the Camera folder in my cell phone. How come I could not access my photos with my computer ? In Volo's Guide it is suggested that a curse could transform someone into a race of 'monstrous origin' (p. 118). Does this apply to the Bestow Curse spell specifically? Note: this is a  (my apologies).  In film / lore, such magic does , ,   - or even  transformative curses.  Points &amp; Concerns:   this is within guidelines for a 'reasonable' curse, i.e. less harm than the listed set Bestow Curse writeup encourages DM-homebrewing &amp;/or interpretation. Great role-play possibility / RAI (hence most myths, fairy tales, cartoons, legends, etc) this could be subject to abuse, temptation to bestow a 'curse' as a blessing / boon one could also change: age, size, weight &amp;/or gender (all great curse ideas... but RAW-legal?)   Summation:  What are the limits of the 3rd lvl / Necromatic Bestow Curse spell for transformation? Enabling tftpd  sudo launchctl load -F /System/Library/LaunchDaemons/tftp.plist   Start:  sudo launchctl start com.apple.tftpd   See logs syslog -w  com.apple.xpc.launchd[1] (com.apple.tftpd[30687]) &lt;Warning&gt;: Service exited with abnormal code: 1   Or with log stream --process tftp --level debug  Filtering the log data using "process BEGINSWITH[cd] "tftp"" Timestamp                       Thread     Type        Activity             PID    TTL 2020-01-21 12:35:44.903240+0200 0x5292a    Activity    0x6afa0              31479  0    tftpd: (libsystem_info.dylib) Retrieve User by Name 2020-01-21 12:35:44.903949+0200 0x5292a    Default     0x0                  31479  0    tftpd: recvfrom: Resource temporarily unavailable   I also tried to run tftpd directly without launchctl sudo /usr/libexec/tftpd -d -i /private/tftpboot/ with the same results.  Earlier I used this configuration and it worked fine. But after some changes (may be OS update) it became broken. Current MacOS version is 10.14.6 (18G2022)    How to fix tftpd?    Appendix. Configuration file /System/Library/LaunchDaemons/tftp.plist has default content:  &lt;?xml version="1.0" encoding="UTF-8"?&gt; &lt;!DOCTYPE plist PUBLIC "-//Apple Computer//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd"&gt; &lt;plist version="1.0"&gt; &lt;dict&gt;     &lt;key&gt;Disabled&lt;/key&gt;     &lt;true/&gt;     &lt;key&gt;Label&lt;/key&gt;     &lt;string&gt;com.apple.tftpd&lt;/string&gt;     &lt;key&gt;ProgramArguments&lt;/key&gt;     &lt;array&gt;         &lt;string&gt;/usr/libexec/tftpd&lt;/string&gt;         &lt;string&gt;-i&lt;/string&gt;         &lt;string&gt;/private/tftpboot&lt;/string&gt;     &lt;/array&gt;     &lt;key&gt;inetdCompatibility&lt;/key&gt;     &lt;dict&gt;         &lt;key&gt;Wait&lt;/key&gt;         &lt;true/&gt;     &lt;/dict&gt;     &lt;key&gt;InitGroups&lt;/key&gt;     &lt;true/&gt;     &lt;key&gt;Sockets&lt;/key&gt;     &lt;dict&gt;         &lt;key&gt;Listeners&lt;/key&gt;         &lt;dict&gt;             &lt;key&gt;SockServiceName&lt;/key&gt;             &lt;string&gt;tftp&lt;/string&gt;             &lt;key&gt;SockType&lt;/key&gt;             &lt;string&gt;dgram&lt;/string&gt;         &lt;/dict&gt;     &lt;/dict&gt; &lt;/dict&gt; &lt;/plist&gt;   Update1 It seems port is in use. Why? Who?  $ sudo lsof -i:69 COMMAND PID USER   FD   TYPE             DEVICE SIZE/OFF NODE NAME launchd   1 root   23u  IPv6 0x55f5dbd8d897c847      0t0  UDP *:tftp launchd   1 root   57u  IPv4 0x55f5dbd8d897e687      0t0  UDP *:tftp launchd   1 root   59u  IPv6 0x55f5dbd8d897c847      0t0  UDP *:tftp launchd   1 root   60u  IPv4 0x55f5dbd8d897e687      0t0  UDP *:tftp   launchd uses UDP port 69. Why?  $ ps -ef | grep tftp 501 94727 48073   0  4:16PM ttys007    0:00.00 grep --color=auto tftp How to show a group is cyclic? One question asking if $\mathbb{Z}^*_{21}$ is cyclic.  I know that the cyclic group must have a generator which can generate all of the elements within the group.  But does this kind of question requires me to exhaustively find out a generator? Or is there any more efficient method to quickly determine if a group is a cyclic group? How can I prove that $\,ax\equiv b\pmod{\! n}\,$ has a unique solution if $\,\gcd(a,n) = 1$?  I suspect that the Bezout identity for $\,\gcd(a,b) = 1$ is the key but I don't see how to proceed. filling an occluded plane with the smallest number of rectangles I've got a specific problem which I'll try to describe as clearly as possible.   I have a defined rectangular region on a cartesian plane, and within that region there are other given rectangular sub-regions that are described in terms of their 4 vertices, ie {(x1, y1), (x1, y2), (x2, y1), (x2, y2)}, so that these regions form 'occlusions' on the plane. These regions don't overlap, but they can form more complex polygons when different-sized rectangles adjacent to each other appear joined.  here's an illustration:   I am interested in the space between these shapes, and how to define the space in the same way the occlusions are defined, that is as a set of rectangles. In particular I want the definition optimised so that the space is described using the smallest possible number of rectangles. For example, an incomplete rendering might look like this:   Can anyone suggest a way forward with this? How can I have the original set of vertices (describing the black rectangles) generate the 'complementary' set of vertices (red rectangles) such that the number of red rectangles is minimal?  I suspect it's a variation of a 'packing problem', but I have a feeling it might be fairly simple...