Matrices: Given $A^{-1} A = I$, prove $A A^{-1} = I$
If $AB = I$ then $BA = I$
What is the formula for percent of frame filled at a specific distance and focal length? I am thinking about buying a new lens specifically to take photos of distant targets (birds, bears, etc). I currently own the Canon 70-300 IS USM, which is a nice lens, but I find that if I want to take a photo of anything further away than about 10 meters, the target does not fill a large amount of the frame (meaning I have to significantly crop in post production).  I am currently looking at the Canon prime 400mm, however I would like to determine at what distance an object will fill a reasonable proportion of the frame.  Is there a formula or rule of thumb I can apply that will help me in this situation? I realise the size of the target I am shooting will play a role here, so if we need to make an assumption about the size of the target please let me know. Where is the rest of Yoda's species? We never seem to see any beings of Yoda's species - except, of course, Yoda himself.  This leads us to the question of where the rest of the li'l critt'rs are. Yoda had to have parents, correct? So what happened to them? Was Yoda, put simply, just one of the last of a dying species? I am trying cmd --> MMC --> Server Manager and then connecting to my Server 2012 no GUI  but i keep getting the following error The WinRM client cannot process the request  I am unable to upload images right now  This is not related to other post "How to Remote Manage Windows Server 2012 in Server Manager?" because title and description of that post is quite different from what i am asking Disable browser 'Save Password' functionality One of the joys of working for a government healthcare agency is having to deal with all of the paranoia around dealing with PHI (Protected Health Information). Don't get me wrong, I'm all for doing everything possible to protect people's personal information (health, financial, surfing habits, etc.), but sometimes people get a little too jumpy.  Case in point: One of our state customers recently found out that the browser provides the handy feature to save your password. We all know that it has been there for a while and is completely optional and is up to the end user to decide whether or not it is a smart decision to use or not. However, there is a bit of an uproar at the moment and we are being demanded to find a way to disable that functionality for our site.  Question: Is there a way for a site to tell the browser not to offer to remember passwords? I've been around web development a long time but don't know that I have come across that before.  Any help is appreciated. I noticed that if I transfer a few large files between two hard drives, it's pretty speedy, at around at least 30&nbsp;MB per second, but if I transfer thousands of files less than 5&nbsp;KB, it is pretty damn too slow.. around 1 to 2&nbsp;MB per second.   Is there a way to speed up the copy/paste process with thousands of small files on Windows 7? Prove that the expression cannot be a power of 2 NOTE: This is a Master Canonical Post for the issues observed on February 25, 2020.  All issues on Feb 25 relating to Python updates not found should be dupe-linked here.  Errors have been seen such as this today, February 25, 2021: Failed to fetch http://security.ubuntu.com/ubuntu/pool/main/p/python2.7/libpython2.7-minimal_2.7.17-1~18.04ubuntu1.3_amd64.deb 404 Not Found  Is there something up with the repositories? I write financial applications where I constantly battle the decision to use a double vs using a decimal.  All of my math works on numbers with no more than 5 decimal places and are not larger than ~100,000. I have a feeling that all of these can be represented as doubles anyways without rounding error, but have never been sure.  I would go ahead and make the switch from decimals to doubles for the obvious speed advantage, except that at the end of the day, I still use the ToString method to transmit prices to exchanges, and need to make sure it always outputs the number I expect. (89.99 instead of 89.99000000001)  Questions:   Is the speed advantage really as large as naive tests suggest? (~100 times) Is there a way to guarantee the output from ToString to be what I want? Is this assured by the fact that my number is always representable?   UPDATE: I have to process ~ 10 billion price updates before my app can run, and I have implemented with decimal right now for the obvious protective reasons, but it takes ~3 hours just to turn on, doubles would dramatically reduce my turn on time. Is there a safe way to do it with doubles? Why did Douglas Adams pick 42 as the ultimate answer? Star Trek often refers to 47 and Star Trek made a sport out of finding real or imagined references to 47. Why did Douglas Adams pick 42 as the ultimate answer in the Hitchhikers guide to the galaxy? What are the colored/highlighted edges in edit mode? Ephemeral error message: "cryptsetup: unknown fstype, bad password or options?" I activated disk encryption for my Ubuntu 15.10 64bit installation and get every time after entering the (correct) password the error message:  cryptsetup: unknown fstype, bad password or options?  But after few seconds (without any further interaction) this error message is replaced by the expected message:  cryptsetup: sda5_crypt set up successfully  That's it and the system is started.  Any suggestions what could be the problem? Is it merely cosmetic or serious? In HLW8012 where should I connect the reference voltage (GND) In the , the both 33nF capacitors , 1K resistor and 0.1uF capacitor are connected to GND. I'm not sure where should I connect those ends of components marked with GND symbol, (To Live, To Natural or the DC GND). I can't read Chinese, I don't really know what everything mean there.  Live directly connects to the shunt resistor and this means that live is also GND (0 volt reference) for the chip? Should I connect those end of components marked as GND symbol to Live wire? Why does ( exit 1 ) not exit the script? I have a script, that does not exit when I want it to.  An example script with the same error is:  #!/bin/bash  function bla() {     return 1 }  bla || ( echo '1' ; exit 1 )  echo '2'   I would assume to see the output:  :~$ ./test.sh 1 :~$   But I actually see:  :~$ ./test.sh 1 2 :~$   Does the () command chaining somehow create a scope? What is exit exiting out of, if not the script? Before Mavericks, I could use /etc/launchd.conf file to change maximum system resource consumption, for example:  limit maxfiles 16384 unlimited limit maxproc 16384 unlimited   It no longer works in Mavericks.  What is the correct way to do it in the recent version of OS X? Is the Ministry of Magic unable to detect illegal time-tampering? Since Hermione already admitted that they were breaking the Law. Would it be safe to conclude that the Ministry cannot instantly (by magic) detect illegal time-tampering?  (Both before and during the events of Prisoner of Azkaban.)     “Hermione,” said Harry suddenly, “what if we — we just run in    there and grab Pettigrew —”       “No!” said Hermione in a terrified whisper. “Don’t you understand?    We’re breaking one of the most important wizarding laws!    Nobody’s supposed to change time, nobody! You heard Dumbledore,    if we’re seen —”       — Prisoner of Azkaban, chapter 21 (Hermione’s Secret) Brightness changes while "Adaptive Brightness" is disabled in Windows 8.1/10 with Intel HD Graphics The brightness changes with the colors displayed on the screen on my Lenovo Thinkpads with Intel HD Graphics. Light colors make the screen brighter, dark make it darker. Same behavior in Windows 8.1 and 10.  The "adaptive brightness" feature in Windows is disabled. Question about preposition -of- Sentence:      A set of deer antlers lying on a flat stone just inside [of?] one cave reveals more recent visitors.   Should we use preposition -of- in that case?  I have checked Corpus of Contemporary American English and have found examples of both usages:        ...to plant another 20 acres of vines, finish building the inside of   an on-site cave, and work on building a retail tasting room (Fiction)   However, opened and remained blue as the inside of an ice cave, paler   than the shell of a robin's egg (Fiction)   It looked like the wall on the inside of a cave: flat and kind of   scale (Fiction)   ...-To protect the inside of the cave. " // He stepped inside, leaving   the gate open.  (Fiction)     Views outside and inside of Wind Cave in South Dakota (Magazine)   Above, Black Hills grasslands and, right, boxwork inside of the cave   (Magazine)   You can see Bill Elliott's great photos of what you can see inside a   cave. (Spoken)   So, when I am asked, what was your feeling inside the cave? (Spoken)   At some point, we found a small river running through the inside the cave (Spoken)   We were gathered around a small fire inside the cave (Fiction)   About 700 feet inside El Sidrn cave, a research team including   Lalueza-Fox excavated 1,700 bones (Academy)      I still cannot come up with a clear pattern why we are using -of- in some cases but are not using it in other. Is it more a question of style rather then a grammar? Or it has something with how speaker/writer sees things around and perceives a reality? Does absence of -of- make things sound weird/wrong for native speaker? Thanks. How to use a C++ string in a structure when malloc()-ing the same structure? Nested sequences of balls in a Banach space This seems to be a fairly easy question but I'm looking for new points of view on it and was wondering if anyone might be able to help.  (By the way- this question does come from home-work, but I've already solved and handed it, and I'm posting this out of interest, so no HW tag.)     Let $B_n=B(x_n,r_n)$ be a sequence of nested closed balls in a Banach space $X$. Prove that $\bigcap_{n=1}^\infty B_n\neq\varnothing$.   As I said before, it should be rather simple. When the radii decrease to 0, it's just a matter of selecting any sequence of points in $B_n$, and it must be Cauchy- and the limit is in the intersection.  My question is what to do when the radii do not decrease to 0? I got some tips about multiplying the balls by a sequence of decreasing scalars, or reducing the radii so that they decrease to 0, but found too many pathological cases for both methods.  Finally- I used a geometric arguemnt (which i've shown to work in any normed space) that if $B(x_1,r_1)\subset B(x_2,r_2)$ then $\| x_1-x_2\|\leq|r_1-r_2|$. This turned out to be some kind of technical catastrophe, but it worked...  Still, if anyone knows of a more elegant solution, I'd love to hear about it. Prove the existence of an analytic function Let $\Delta=\{z\in \mathbb C: |z| &lt; 1\}$. Let $f: \Delta \rightarrow \mathbb C$ be a one-to-one analytic function fixing the origin. Prove that there is a one-to-one analytic function $g: \Delta \rightarrow \mathbb C$ such that $[g(z)]^2=f(z^2)$. Further, show that such a function is odd.  The only idea which comes to my mind is the following (not sure whether it is the right direction though). Let $h: \Delta \rightarrow \Delta \rightarrow \mathbb C,\ z\mapsto z^2\mapsto f(z^2)$. Note that $h'(0)=0,\ h''(z)=2f'(z^2)+4z^3f''(z^2)$; since $f$ is one-to-one and analytic, its derivative never vanishes, so $h''(0)\ne 0$. So $h$ is an analytic function of $\Delta$ with a zero of order two at the origin. Then there is an analytic function $g$ defined in an open disc about the origin with a simple zero at the origin such that $[g(z)]^2=f(z^2)$.   This is the only thing I came up with, but it seems to be a little bit irrelevant since I don't see any way of extending this $g$ to the whole $\Delta$, and also the injectivity and oddness of such a $g$ are unclear.