I've read that function composition "generally does not commute."  Not counting compositions involving the identity function, and compositions of a function and its inverse, are there examples of functions on the reals (for example) $f, g$ where $fg = gf$ outside of these cases?
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?
It seems as if a shift occurred and the descriptive "Oriental" was replaced by "Asian" as the accepted term in polite society — what caused this shift? Drop rates for set / legendary items Just wondering what the drop rates were for Set / Legendary items, or is it totally random? I imagine they would increase for later acts in Inferno. My monk is comfortable in Act 3 in Inferno, but I've only found one set item, so MF will increase chances for them? Create a table with LaTeX I need to copy a table from a pdf file to my LaTeX document. I know I have to do it by hand but still don't understand how to create a table with LaTeX. Why are neural networks becoming deeper, but not wider? In recent years, convolutional neural networks (or perhaps deep neural networks in general) have become deeper and deeper, with state-of-the-art networks going from 7 layers () to 1000 layers ( in the space of 4 years. The reason behind the boost in performance from a deeper network, is that a more complex, non-linear function can be learned. Given sufficient training data, this enables the networks to more easily discriminate between different classes.  However, the trend seems to not have followed with the number of parameters in each layer. For example, the number of feature maps in the convolutional layers, or the number of nodes in the fully-connected layers, has remained roughly the same and is still relatively small in magnitude, despite the large increase in the number of layers. From my intuition though, it would seem that increasing the number of parameters per layer would give each layer a richer source of data from which to learn its non-linear function; but this idea seems to have been overlooked in favour of simply adding more layers, each with a small number of parameters.  So whilst networks have become "deeper", they have not become "wider". Why is this? What are the most accepted ways to visualize the results of an independent two sample t-test? Is a numeric table more often used or some sort of plot? The goal is for a casual observer to look at the figure and immediately see that they are probably from two different populations. Performing multiple layer intersection in QGIS? Fighting a black hole: Could a strong spherical shell inside an event horizon resist falling in to the singularity? As a thought experiment imagine an incredibly strong spherical shell with a diameter a bit smaller than the event horizon of a particular large black hole.  The shell is split into two hemispheres, and the two hemispheres are released from either side of the black hole such that they will meet and join inside the event horizon, forming a stationary uniform shell around the singularity. Of course, we consider that the shell is flawless, and exactly centred around a stationary singularity - clearly this is an unstable equilibrium but we imagine it is possible.  Is is possible that the structural forces provided by such a shell could be strong enough to resist the extreme gravity conditions inside the event horizon, and hence avoid being sucked into the singularity?      (For the purposes of this question we can consider taking the shell to extreme levels of strength, e.g. consider that the shell is made from a theoretical maximally densely packed neutronium-like material that is strongly resistant to any further compression due to the Pauli exclusion principle). Sql Server 'Saving changes is not permitted' error ► Prevent saving changes that require table re-creation How to run ubuntu/xubuntu in a complete non-gui mode Send mail without MFMailComposeViewController Is it possible for information to be transmitted faster than light by using a rigid pole? Countability of disjoint intervals According this problem/solution set from an MIT class (http://ocw.mit.edu/courses/mathematics/18-100c-analysis-i-spring-2006/exams/exam1_sol.pdf), the assertion:  "Every collection of disjoint intervals in R is countable."  is True, because "every interval contains a rational number", and the rationals are countable.  It seems to me this should be False, with possible counterexample:  { [x,x] | x is an element of R}  ie the set of all singelton intervals on R. Why isn't this a valid counterexample? How do I bake multiple texture maps into 1 UV, In Blender Cycles? Using:  Blender V2.79 Cycles Render Engine  After creating a complex interior scene consisting of 3 different texture types, each having 3 associated maps (Diffuse, Gloss, Normal) I noticed the texture and baking options will not successfully combine to create a texture for the exported mesh once I bring it into Unity 3D. The mesh is a single object with the different textures assigned to specified faces.  I've played around with a default cube mesh for testing purposes, using the same 3 textures split among the 6 sides, but have only been able to successfully pull over the color info by baking a diffuse map to my UV and saving the image. When using the other baking options, Blender does not successfully create a map for either my Glossy textures, nor my normal map. Even using the "Combined" baking option, I am provided a flat dull diffuse image for my mesh, no gloss or normal info is saved.  My Node process for textures is displayed here  Here is the result of baking my textures as a Normal  The result of adding other textures made from Blender's baking, results in flat gloss at best.  What can I do differently in Blender, to have my assigned UV Diffuse, Normal, and Gloss maps all appear in a single object in Unity? What are 'return' pins? On a VGA connector, there are numerous 'ground' pins. Three of these are Red return, Green return and Blue return; But why does each color component have its own ground pin? Proof, is $\forall x : (P(x) \lor Q(x)) \Leftrightarrow \forall x : P(x) \lor \forall x : Q(x)$? I have to prove if the following statement is true or false $$\forall x : (P(x) \lor Q(x)) \Leftrightarrow \forall x : P(x) \lor \forall x : Q(x)$$  I understand the first statement as, for every $x$, at least one of the functions $P$, $Q$ is true. For the second statement, at least one of the functions $P$, $Q$ is true for every $x$. Is this correct? When yes, how could I prove this? Using the distributive law will clearly render me the wrong result. How should I setup separate MX records for a subdomain? Let's say I have a domain that I run a web application on, for example cranketywidgets.com, and I'm using  for handling email for people working on that domain, for example, support@ cranketywidgets.com, jane@cranketywidgets.com, joe@cranketywidgets.com and so on.  Google's own mail services aren't always the best for sending automated reminder emails, comment notifications and so on, so the current solution I plan to pursue is to create a separate subdomain called mailer.cranketywidgets.com, run a mail server off it, and create a few accounts specifically for sending these kinds of emails.  What should the MX records and A records look like here for this?  I'm somewhat confused by the fact that MX records can be names, but that they must eventually resolve to an A record. What should the records look like here?  cranketywidgets.com - A record to actual server like 10.24.233.214  cranketywidgets.com - MX records for Google's email applications  mailer.cranketywidgets.com - MX name pointing to server's IP address  I would greatly appeciate some help on this - the answer seems like it'll be obvious, but email spam is a difficult problem to solve. Arbitrary products of quasi-coherent sheaves? When traveling to a country with a different currency, how should you take your money? Sarrus Rule with LaTeX Is there a way to create a new command with 9 arguments ({a_{11}},{a_{12}},{a_{13}},{a_{21}},{a_{22}},{a_{23}},{a_{31}},{a_{32}},{a_{33}}) that has as output a matrix with the Sarrus Rule? For instance: to obtain     just use the command that we will call sarrus:  \sarrus{-k}{k-1}{1}{0}{k-1}{k}{2}{0}{1}.   If it is possible i want to use the command sarrus in math mode just like matrix for istance. Say person A and person B are having a discussion/argument.   Person A tells person B that his views are arrogant, or closed-minded, or some other derogatory adjective. But in fact it is person A's views that are arrogant, closed-minded, etc. Person B's views are usually the exact opposite of what person A is accusing him of.  This sort of situation happens a lot when discussing politics or religion, for example.  What do you call person A, for calling person B something that he himself is?  I guess, for the most part, person A doesn't actually realize that he is the one that the adjective applies to, and not person B. Maybe there is a different name if person A knows the adjective also applies to him.