Functions whose derivative is the inverse of that function
Function whose inverse is also its derivative?
A site URL changed and now its icon in SEDE is broken Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$? I was wondering if there is a name for a function that satisfies the conditions  $f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \cdot f(y)$?  Thanks and regards! Difference between static class and singleton pattern? What is a nicer, less immature saying?  I love using 'A Freudian slip', but that is only applicable in certain situations. Moment of inertia of a hollow sphere wrt the centre? I've been trying to compute the moment of inertia of a uniform hollow sphere (thin walled) wrt the centre, but I'm not quite sure what was wrong with my initial attempt (I've come to the correct answer now with a different method). Ok, here was my first method:  Consider a uniform hollow sphere of radius $R$ and mass $M$. On the hollow sphere, consider a concentric ring of radius $r$ and thickness $\text{d}x$. The mass of the ring is therefore $\text{d}m = \frac{M}{4\pi R^2}\cdot 2\pi r\cdot\text{d}x$. Now, use $r^2 = R^2 - x^2:$ $$\text{d}m = \frac{M}{4\pi R^2}\cdot 2\pi \left(R^2 - x^2 \right)^{1/2}\text{d}x$$ and the moment of inertia of a ring wrt the centre is $I = MR^2$, therefore: $$\text{d}I = \text{d}m\cdot r^2 = \frac{M}{4\pi R^2}\cdot 2\pi\left(R^2 - x^2\right)^{3/2}\text{d}x $$ Integrating to get the total moment of inertia: $$I = \int_{-R}^{R} \frac{M}{4\pi R^2} \cdot 2\pi\cdot \left(R^2 - x^2\right)^{3/2}\ \text{d}x = \frac{3MR^2 \pi}{16}$$  which obviously isn't correct as the real moment of inertia wrt the centre is $\frac{2MR^2}{3}$.   What was wrong with this method? Was it how I constructed the element? Any help would be appreciated, thanks very much. Is there a known well ordering of the reals? So, from what I understand, the axiom of choice is equivalent to the claim that every set can be . A set is well ordered by a relation, $R$ , if every subset has a least element. My question is: Has anyone constructed a well ordering on the reals?  First, I was going to ask this question about the rationals, but then I realised that if you pick your favourite bijection between rationals and the integers, this determines a well ordering on the rationals through the natural well order on $\mathbb{Z}$ . So it's not the denseness of the reals that makes it hard to well order them. So is it just the size of $\mathbb{R}$ that makes it difficult to find a well order for it? Why should that be?  To reiterate:   Is there a known well order on the Reals? If there is, does a similar construction work for larger cardinalities? Is there a largest cardinality for which the construction works? Frequent Write to Non-volatile Memory Export File Dialog in Blender 2.80 How can I get bring up a popup file browser in Blender 2.80 invoked by a button click in my addon, to define the name and location of a file to be exported?  In Blender 2.7x, I did this via the context window manager in the invoke function, but it is very possible that this hasn't been the preferred method for years, as I'm not finding useful documentation when I search for how to update this.  class MYTOOL_OT_write_data(bpy.types.Operator):     bl_idname = "mytool.write_data"     bl_label = "Write data file"      directory: bpy.props.StringProperty(subtype="FILE_PATH")     filename: bpy.props.StringProperty(subtype="FILE_NAME")      def invoke(self, context, event):  # 2.7x method         WindowManager = context.window_manager         context.window_manager.FILEBROWSERect_add(self)  # &lt;-- error here         return{"RUNNING_MODAL"}      def execute(self, context):         directory = self.directory         filename = self.filename         # Write csv data file using normal python functionality         return{'FINISHED'}   In Blender 2.80, I am getting the error "AttributeError: 'WindowManager' object has no attribute 'FILEBROWSERect_add'".  Thanks for your help.    UPDATED SOLUTION based on brockmann's solution below.  The trick is to use the ExportHelper util (which existed long before Blender 2.80):  from bpy_extras.io_utils import ExportHelper  class MYTOOL_OT_write_data(bpy.types.Operator, ExportHelper):     bl_idname = "mytool.write_data"     bl_label = "Write data file"      filename_ext = ".csv"  # ExportHelper mixin class uses this     def execute(self, context):         filepath = self.filepath         # f = open(filepath, 'w')         # f.write(stuff)         # f.close()         return{'FINISHED'} Are used Schengen/UK visas acceptable for Turkish e-visa Dollar Sign corrupts question In  my question () contains a "$" in a quoted section of the question.  Originally I had a second "$" in a second quoted  section of the same question.    However when I did this the second $ seemed to trigger a style that made the text italic and lost spacing between words.  I took out the second "$" and instead emphasized it with the word "dollars" which solved the immediate problem.  However, in the listing of the question, (not the question itself) the same thing happens after the first "$".  The summary shows the text without spaces between words and in italic.  Seems like a bug.  Below is a verbatim copy/paste of the original issue (two dollar signs):    I was able to find a small ESC for about $12 off of ebay.  If you were designing a robot, would you see that and think?       $12 bucks for an ESC that connects to simple pulse-wave interface -   sign me up!   Or would you think:     $12 just to control a motor? I could throw together an H-bridge for   $0.50 and be done with it.   My robot in particular actually has two motors and therefor $24 to control the two of them.  But the interface is really easy (plus has the added advantage of being R/C vs computer controlled with a simple change of connectors.  Which way would you go? Is it possible to produce such design hints (on elements padding, spacing, width, height) automatically in photoshop? Is there any plugin that can potentially do this? Approximating $\arctan x$ for large $|x|$ Copy Physics To Multiple Objects Using the cell fracture addon, I created many objects from my mesh. I want to have a rigid body physics on all of them. How would I copy one of the fractured pieces physics to all the pieces? I tired ctrl L and the copy attributes addon, but none of them have the option. Any help is great. Using some matlab (I know it's cheating) I found that: $$k=0 =&gt; Result=1$$ $$k=1 =&gt; Result=1$$ $$k=2 =&gt; Result=2$$ $$k=3 =&gt; Result=5$$ $$k=4 =&gt; Result=15$$ $$k=5 =&gt; Result=52$$ $$k=6 =&gt; Result=203$$  This sequence is puzzling... Who is doing the normalization of wave function in the time evolution of wave function? In the Schrödinger equation, at any given time $t$ we should jointly add another sub equation, like $$||\psi_t(x)|| = 1$$ where $\psi_t(x) = \Psi(x,t)$, and then try to solve the two equations simultaneously. Why not? I know it does not yield, but I am always baffled, who is doing the normalization of the wave function? Observers, the system, the measuring process, God? No space following macro without argument With a macro such as:  \def\naive{na\"{\i}ve}   I find that the space which I would hope to follow it, is absent. So, with  the \naive approach   I get     the naïveapproach   in the output. How do I bring back the space? Normal subgroups of infinite symmetric group I recently took a course on group theory, which mentioned that the following proposition is equivalent to the continuum hypothesis: "The infinite symmetric group (i.e. the group of permutations on the set $\mathbb{N}$) has exactly 4 normal subgroups." Does anyone have any references or explanation for this? Mount error: "unknown filesystem type 'exfat'" When trying to mount an exfat filesystem, I get the following error:  Error mounting /dev/sda6 at /media/gkp/Backup: Command-line  `mount -t "exfat" -o "uhelper=udisks2,nodev,nosuid" "/dev/sda6" "/media/gkp/Backup"'  exited with non-zero exit status 32: mount: unknown filesystem type 'exfat'   Exfat is used on some USB sticks and camera sd cards. What can I do to mount this type of filesystem? GPG can't check signature Definite article before name of languages Should I use the definite article before name of languages?   (the?) Persian language (the?) German language (the?) Arabic language (the?) Russian language (the?) English language   Some language name may also refer to other things, so you need language word after their name.