What are the functions that are their own inverse?  (thus functions where $ f(f(x)) = x $ for a large domain)  I always thought there were only 4:  $f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ f(x) = \frac {-1}{x} $   Later I heard about a fifth one     $$f(x) = \ln\left(\frac {e^x+1}{e^x-1}\right) $$ (for $x &gt; 0$)  This made me wonder are there more? What are conditions that apply to all these functions to get more, etc.
I can think of several examples of functions such that twice application of the function is equivalent to no application of it.   Additive inverse Multiplicative inverse Fourier transform Complex conjugation Any group built up from $\mathbb{Z}_2$, applying (one of) the $\mathbb{Z}_2$ parts' operation.   "Idempotent" came to mind, but that's wrong. It means $f(f(x)) = f(x)$, not $f(f(x))=x$.  What is the word for this "flip-flop" property?
What is the name of this kind of character illustration? I have seen lots of UI designer nowadays including this kind of illustration i want to know the name of this style so that i can also learn and try this in my designs. What does it mean when multimeter accuracy is marked as: ±0,03%+10Digit? I have a digital multimeter and its accuracy for VDC is marked like this:      ±0,03%+10Digit   This multimeter has maximum display of 80000. So in the 80 V range it can show for example 79.999V.  0.03% of 80V is 0.024V - that is clear for me. But what does the +10Digit mean?  The device in question in . How to compile .c file with OpenSSL includes? Why on the first page Google says there are thousands of results but on the last page there are less than a hundred? Default implementation for Object.GetHashCode() I designed a poster in CorelDraw X5 for printing purposes. I was told by the Printshop guy to use colors only in CMYK. For the most part I did, I don't know why I missed it but I used one color in RGB. I liked this color so much that I've used it at many places. So, current situation is that some colors are in RGB and some are in CMYK. So, I want to convert all RGB to CMYK and thought the problem would be solved. But now I'm actually facing a lot of trouble. The colors are not at all same. I hate the colors that it is showing me.  This is the color I used in RGB:    When I changed the "Model" to CMYK, the color it shows is really different. As you can see in the picture below, it's too dull:    The RGB values of both colors are not the same. What can I do to get the same color in CMYK? I did a little bit of reading here and there. They say, that you can't get same. In that case, what is the closest color that I can get? cd to previous directory without echo of directory name This  gave me cd - as shorthand to change to my previous directory in Bash. It does have a major problem: it prints the directory it changes to, wasting my screen space, as that directory is then repeated in the prompt on the next line.  I know I can do cd $OLDPWD, also mentioned in that answer, and cd - &gt;/dev/null but that is much more typing.   In the bash man page it says:     ... or if - is the first argument, and the directory change is successful, the absolute path name of the  new  working  directory  is written to the standard output.   How can I make - not to be the first argument? By using any of the cd optional arguments? Any other way to suppress the echo or minimise typing without the need for defining functions/aliases on every machine I work with? What's the difference between the 'ref' and 'out' keywords? I'm creating a function where I need to pass an object so that it can be modified by the function. What is the difference between:  public void myFunction(ref MyClass someClass)   and  public void myFunction(out MyClass someClass)   Which should I use and why? Closures in a for loop Why do we have two of some organs, but not all? We have two eyes, but we don't have two hearts. Why do humans have two of some organs, but not all? How do I make Chromium use Flash from Google Chrome? Now, that Adobe has stopped issuing new releases for Linux desktops, the only option for users is to use Google Chrome, for those who want to use the latest Flash updates. I use Chromium. Is there any way to get Chromium use the Flash from Google Chrome? The reason I as this question is that I have noticed both these browsers run in the background, so technically, the browsers' plugins are available to use. RSA: Fast factorization of N if d and e are known note: this question has been updated to reflect that we are currently using MySQL, having done so, I would like to see a how much easier it would be if we switched to a CTE-supporting database.  I have a self-referencing table with a primary key, id and a foreign key parent_id.  +------------+--------------+------+-----+---------+----------------+ | Field      | Type         | Null | Key | Default | Extra          | +------------+--------------+------+-----+---------+----------------+ | id         | int(11)      | NO   | PRI | NULL    | auto_increment |  | parent_id  | int(11)      | YES  |     | NULL    |                |  | name       | varchar(255) | YES  |     | NULL    |                |  | notes      | text         | YES  |     | NULL    |                |  +------------+--------------+------+-----+---------+----------------+   Given a name, how can I query the top-level parent?  Given a name, how can I query all of the id's associated with a record of name = 'foo'?  context: I am not a dba, but am planning to ask a dba to implement this type of hierarchical structure and would like to test some queries. The motivation for doing so is described by .    Here is an example of the relationships among ids in the table:    -- ----------------------------------------------------- -- Create a new database called 'testdb' -- ----------------------------------------------------- SET @OLD_UNIQUE_CHECKS=@@UNIQUE_CHECKS, UNIQUE_CHECKS=0; SET @OLD_FOREIGN_KEY_CHECKS=@@FOREIGN_KEY_CHECKS, FOREIGN_KEY_CHECKS=0; SET @OLD_SQL_MODE=@@SQL_MODE, SQL_MODE='TRADITIONAL';  CREATE SCHEMA IF NOT EXISTS `testdb` DEFAULT CHARACTER SET latin1 COLLATE latin1_swedish_ci ; USE `testdb` ;  -- ----------------------------------------------------- -- Table `testdb`.`observations` -- ----------------------------------------------------- CREATE  TABLE IF NOT EXISTS `testdb`.`observations` (   `id` INT NOT NULL ,   `parent_id` INT NULL ,   `name` VARCHAR(45) NULL ,   PRIMARY KEY (`id`) ) ENGINE = InnoDB;  SET SQL_MODE=@OLD_SQL_MODE; SET FOREIGN_KEY_CHECKS=@OLD_FOREIGN_KEY_CHECKS; SET UNIQUE_CHECKS=@OLD_UNIQUE_CHECKS;  -- ----------------------------------------------------- -- Add Example Data Set -- -----------------------------------------------------   INSERT INTO observations VALUES (1,3), (2,5), (3,NULL), (4,10),     (5,NULL), (6,1), (7,5), (8,10), (9,10), (10,3); Are proofs by contradiction really logical? What is the average & max temperature of an airplane cargo hold? I want to take some chocolates to offer in my luggage. Chocolates are a bit sensitive to heat, though. What is the average temperature inside a cargo hold? 'Average by' itself can be misleading. What is the max temperature there?  I don't know much about airplane mechanics but I assume they have some cooling systems like any other engine. Such systems, depending on where they are placed, might influence temperature in the cargo hold. I also fear temperatures in the ground while exterior temperatures are higher, and luggage is being put inside. What are max temperatures near the airplane in such situation? I'm trying to install the Jupiter power settings app but, I read it is no longer compatible with the new kernel. Is there a similar program or a way to manage the power settings on 13.04? I am running Ubuntu 20.04 an  with an NVIDIA GeForce GTX 1660TI GPU. According to , the driver version for this GPU is 430. When I first installed Ubuntu, I installed 440. I'm now trying to correct that and install 430. I ran sudo apt-get install nvidia-driver-430, and then did a reboot. When I examined nvidia-smi, though, it still showed the 440: +-----------------------------------------------------------------------------+ | NVIDIA-SMI 440.100      Driver Version: 440.100      CUDA Version: 10.2     | |-------------------------------+----------------------+----------------------+ | GPU  Name        Persistence-M| Bus-Id        Disp.A | Volatile Uncorr. ECC | | Fan  Temp  Perf  Pwr:Usage/Cap|         Memory-Usage | GPU-Util  Compute M. | |===============================+======================+======================| |   0  GeForce GTX 166...  Off  | 00000000:01:00.0 Off |                  N/A | | N/A   55C    P0     9W /  N/A |      0MiB /  5944MiB |      0%      Default | +-------------------------------+----------------------+----------------------+  What is the appropriate way of updating this driver? Our GM won't let us affect the story; the more I play the less fun I have. What do I do? I was wondering if anyone could give me some advice or anything on a GM situation.   My current GM is running a campaign and it is very video game/anime like. It started out super fun and the deeper we get the less fun I have. The reason is, it feels like GM is telling us a story but we cannot interact or change anything during his, "cut scenes". No matter what any player does he just says, "No it didn't work," or, "Yeah you do that. See if you roll a 20." We also can't damage any boss he has until we reach the point in the story that he has planned, so our interactions we have with villains don't matter.  I feel like he is making us sit there and listen to an anime he has come up with in his head. Since the players can't change or react to the key situations. I find my self sitting there thinking, why would I react, it doesn't matter.  It's not just me either. Another player feels the same way. I am going to talk to my other fellow players soon about it and see how they feel.  Side note: the GM as a person is very defensive and I am afraid to confront him about it because I know he will throw a fit. We have also caught him straight lying about abilities and stats on his own PC in other campaigns, and he doesn't take to kindly when you call him out.  He is new to GMing. This is only the second game I have played with him as a GM. The first one had one session the he just never wrote more for.  Is there anything I can do to maybe get him to more lenient with his story? I really enjoy the story for the most part and if he would just let our characters do things and give us a chance at stopping his villians or letting us change the story based on our actions, everything would be cool. What do you call a person who does/thinks the opposite of the crowd for just that reason? If I know $\| f_n - f \|_{L^p(\mathbb{R})} \to 0$ as $n \to \infty$, do I know that $\lim_{n \to \infty}f_n(x) = f(x)$ for almost every $x$?