Are there two real-valued functions defined on the same subset of $\mathbb{R}$ that commute with each other but are not inverses of each other?  (After several responses, I have to make an edit to my post.  Neither function should be the identity function nor the zero function.  The two functions should not be the same function.)
Today I was thinking about composition of functions.  It has nice properties, its always associative, there is an identity, and if we restrict to bijective functions then we have an inverse.  But then I thought about commutativity.  My first intuition was that bijective self maps of a space should commute but then I saw some counter-examples. The symmetric group is only abelian if $n \le 2$ so clearly there need to be more restrictions on functions than bijectivity for them to commute.  The only examples I could think of were boring things like multiplying by a constant or maximal tori of groups like $O(n)$ (maybe less boring).  My question: In a euclidean space, what are (edit) some nice characterizations of sets of functions that commute? What about in a more general space?   Bonus: Is this notion of commutativity important anywhere in analysis?
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? Why are map tiles being repeated? Using MapServer and Openlayers AutoHotKey: Calling one script from another script Finding plane equation given two lines Determine an equation of the plane containing the lines $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-5}{6}$; $r=\lt1,-1,5\gt+t\lt1,1,-3\gt$.   I calculated the cross product between the directional vector of both lines to find the normal vector $n$, but when I looked for an intersection point $r_0$ to apply the formula: $\lt r-r_0\gt \bullet \ \ n$, I did not find any.  Can I use the point $\lt1,-1,5\gt$ given in the line $r$? Or is it not possible to find an equation of the plane containing two lines that do not intersect? 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. I played 1st and 2nd edition AD&amp;D a lot back in the day. But when 3rd edition came out, I dropped out of the hobby. Partly I just couldn't be bothered to re-learn the rules. The conversion systems, previews and material for 3rd edition I was seeing in Dragon and Dungeon magazines made it look like a very major change.  I recently picked up the new 5th edition starter set on a whim. Obviously the starter set rules are very stripped down in comparison to the full rules, but nevertheless, I was amazed by how comfortable and familiar the system felt. The only thing that felt like a truly radical change was the way things like thief skills and the like were now handled with a d20 instead of a percentage system.  There's plenty of information online (and here) comparing editions of the game in an incremental manner. But I'd be interested in a breakdown of how 2nd and 5th edition compare directly, specifically looking at whether there are broad areas of similarity. Why is compactness in logic called compactness? In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set.   Most logic texts either don't explain the terminology, or allude to the topological property of compactness. I see an analogy as, given a topological space X and a subset of it S, S is compact iff for every open cover of S, there is a finite subcover of S. But, it doesn't seem strong enough to justify the terminology.   Is there more to the choice of the terminology in logic than this analogy? How are strings passed in .NET? When does the amount of skew or prevalence of outliers make the median preferable to the mean? I know that "deviations in the data are devil", and when the distribution is highly skewed, it is better to consider median as average rather than mean, but how to decide these hard-limits.  For example:   CASE 1:   Assume X = 10,20,30,40,50,60,70 In this case, I think that it is better to use mean and that it will give very accurate results.  CASE 2:   Assume X = 10,20,30,40,50,60,70,7000 In this case, I think that it is better to use median instead of using the mean.  CASE 3:    Assume X = 10,20,30,400,500,600,700 In this case, I think it is better to use IQR (Inter Quartile Range)    But I'm stuck with how to decide these hard-limits i.e. which to use in which condition, in general.   I've found a tool working on subjected principle, which takes context-less sample-distribution as input and determines whether mean is close/moderate or against the null-hypothesis.  Find References:-     (For Mean Test)  (For Median Test)   What I'm really looking is a good answer which states how to derive these conclusions. Recursively rename files and directories Using Boolean parameters in Python Script Tool of ArcPy? I'm using ArcGIS for Desktop 10.3.0.4322 I have a script tool which requires a parameter (folder path to a gdB, via GetParameterAsText) so that a later process can be saved there. The thing I want to do is make an Optional Boolean parameter, that when checked, automatically fills in the folder path gdB parameter. The Boolean text asks the user, &quot;Use default workspace?&quot;, to which if they check the box (boolean == True), then the folder path parameter will be automatically defaulted (saving the user a small amount of time). I have already created the Boolean parameter in the script tool, I am however unaware of how to &quot;fill in&quot; the required parameter based on the user's Boolean. Code Snippet: # Ask user to select the Geodatabase workspace to use for data output userWorkspace = arcpy.GetParameterAsText(0)  # Ask user if they want to use default workspace (boolean) useDefault = arcpy.GetParameterAsText(1)  # If user wants default workspace, then give it to the required parameter (0) if useDefault == True:  arcpy.env.workspace = path else:  # Set workspace environment based upon user's choice  arcpy.env.workspace = userWorkspace How can I rename many files at once? I need to rename a bunch of files in order to fix their sort order, therefore I need to be able to do a sort of "find and replace" so I can replace chunks of the filenames at a time.  What tools should I use? I prefer GUI but a command line tool recommendation would also be fine. How is the passage of time handled in Pokemon? I stopped following the Pokemon anime a while ago, but I know the series has had Ash go through many regions, including retreading at least one of them.  When Ash was about to get his eight badge, Misty mentioned that they'd been on the road for a year at that point.  I saw some recent episodes, and Ash looks nearly identical to when he first started as a trainer.  Normally I'd attribute that to cartoon sitcom logic, where the characters never age; but in a series with continuity and references to earlier seasons, this makes no sense.  What's the official word on a timeline for the Pokemon anime? I need help to solve following problem from Rudin's Mathematical analysis book:    Convergence of the series $\sum a_n$  implies the convergence of $\sum \dfrac{\sqrt {a_n}}{n}$, if $a_n&gt;0$  I tried to construct a suitable convergence sequence $b_n$ such that $\sum b_n$ converges and $a_n \leq b_n$ but, I am not able to find such sequence $b_n$ .  Thanks for the help and sugestions. Cannot open adobe reader on Ubuntu 12.10 64bits If $\mathbf A$ and $\mathbf B$ are Hermitian, when does $\det(\mathbf A-\lambda\mathbf B)=0$ have only real roots? Let $\mathbf A, \mathbf B$ be Hermitian matrices of the same size. What is the characterization of $\mathbf A, \mathbf B$ such that $p(\lambda)=\det(\mathbf A-\lambda\mathbf B)=0$ has only real roots?   If $\mathbf B$ is positive definite (I corrected this), it is easy to see $p(\lambda)$ has only real roots. I guess it is too difficult a question to ask about the cardinality of $S_{\mathbb{N}}$ so I would like to ask whether it is countable or not.  I tried to prove it is uncountable somewhat mimicking the Cantor's diagonal argument but failed. What is the difference between the following class methods?  Is it that one is static and the other is not?  class Test(object):   def method_one(self):     print "Called method_one"    def method_two():     print "Called method_two"  a_test = Test() a_test.method_one() a_test.method_two() What is the purpose of Jordan Canonical Form? l, ls & la -- what are the differences, and are there more of these commands?