Does this choice require Axiom of Choice? This theorem is from Munkres Topology 2nd ed.:       In the last para, it is said that 'choose an element' $n$ of $D$. Why we do not need axiom of choice here? (Here $D$ may not be finite.)
How do I choose an element from a non-empty set? Suppose I have a non-empty set $A$.     How do I choose an element $x\in A$?   More precisely, I believe I would like to find a formula $P(x,y)$ of ZF such that for every non-empty set $y$ there is exactly one $x\in y$ such that $P(x,y)$ is true.  I have thought about this for a while without success and I am beginning to doubt such a formula is even possible. But in mathematical proofs it is quite common to "choose a fixed element" from a non-empty set. So how exactly do we achieve such a thing? Am I missing something?  Thank you in advance.
Can someone show me a straightforward proof that  $\lim_{m\to\infty}(1+\frac{r}{m})^{mt}=e^{rt}$  Thanks! Relation between rank and number of distinct eigenvalues of a matrix Let $T : V\to V$ be a linear transformation such that $\dim\operatorname{Range}(T)=k\leq n$, where $n=\dim V$. Show that $T$ can have at most $k+1$ distinct eigenvalues.   I can realize that the rank  will correspond to the number of non-zero eigenvalues (counted up to multiplicity) and the nullity will correspond to the 0 eigenvalue (counted up to multiplicity), but I cannot design an analytical proof of this.  Thanks for any help . I just noticed that the  user edited a ~1 year old  of mine. As you can see, the method call was changed from getSfGuardUser to getGuardUser:    Now I checked the documentation and it seems that the edit is correct.  But how did Community know about that? As far as I know, it is an automated process. Does it have a reference of all possible APIs of languages/frameworks (which seems very unlikely) or is there a routine admins can trigger, for replacing content? Or is it a secret? What I am working with: I am running my self-made Arduino board (in the sense that I use the Arduino bootloader and code editor) at 3.3V, and powered from a Lithium ion battery, which is USB-charged by a corresponding Microchip charger IC.  What I am trying to achieve: I want to measure battery capacity once every minute or so. I have an LCD attached, so the idea is that the overall setup lets me know how the battery is doing at a given moment. The datasheet of the battery has a voltage versus discharge-level curve, and so by measuring the voltage of the battery, I can estimate the remaining capacity (very roughly but enough for me!).  What I did:   (EDIT: Resistor values updated and P-MOSFET switch added based on @stevenvh and @Jonny's suggestions).  I connected a voltage divider from the battery V_plus, with the larger "portion" going to an analog-read-pin (i.e., ADC) on the Arduino/Atmega chip. The divider is 33 KOhm-to-10 KOhm, thus allowing measurement up to 4.1 Volts maximum of the Li-ion battery from my 3.3V level microcontroller. Also, using one of the I/O pins connected to an n-channel MOSFET, I can switch the current through the divider only when I need the measurement. Here is a rough schematic (updated for a 2nd time based on suggestions of @stevenvh and @Nick):     My question:   How is my current setup? My only constraints are: (1) I would like to make a rough measurement of the battery capacity based on the voltage reading, as described above. (2) I would like to prevent the voltage divider from interfering with my charging IC's reading of battery presence (in my original setup, the divider sometimes caused the IC to misread presence even when the battery was absent). "The pair was ..." or "the pair were ..." What is the difference between Python's list methods append and extend? Synaptics driver doesn't load sometimes How to create a circular shaped grid, similar to a manhole cover I’m trying to create a circular shaped grid, similar to the one in the pic. That’s my first attempt, created by arranging several cylinders with an Array modifier and then cutting the holes via the Boolean modifier.  Now you will probably notice that it looks not elegant in any way. I’d like to have the holes perfectly arranged and in a correct order, so that there are no half holes at the border of the mesh, for example.  How would you do this?   I tried working with Dupliverts but that didn’t work out well. how much space does a given package, and dependencies, cost? How to find the id of the last added feature using pyQGIS? Is it possible to programmatically (in python) access the attribute info of the last added feature in QGIS 2.0 before edits are saved? I'm failing to find a way to get the ID and subsequently other attributes of my feature of interest (the one most recently added).  idx = layer.fieldNameIndex('myField') myList = [] lastEdited = ID THAT I'D LIKE TO RETRIEVE for feature in layer.getFeatures():     if feature.id() == lastEdited:         print feature.attributes()[idx]   From this I'd like to get 'myField' of my most recent addition.   UPDATE: Following lead of this , I was able to achieve my goals via the Python console, but am failing to get it to work for my plugin.  def updateTableWidget(self, fid):     idx = layer.fieldNameIndex('myField')     myList = []     for feature in layer.getFeatures():         if feature.id() == fid:             myList.append(feature.attributes()[idx])   Via the python console, I was able to connect to my layer like so:  layer.featureAdded.connect(updateTableWidget)   After adding a QgsMessageLog to my function, the attributes were correctly identified. I now have a beginner python issue, however. Calling the function in my plugin as shown below gives me TypeError: updateTableWidget() takes exactly 2 arguments (1 given):  self.updateTableWidget()   The function was called from feature form python file. Is soaking beans 24 hours unrefrigerated safe? How do the likely/unlikely macros in the Linux kernel work and what is their benefit? What happens to the energy when waves perfectly cancel each other? What happens to the energy when waves perfectly cancel each other (destructive interference)? It appears that the energy "disappear" but the law of conservation of energy states that it can't be destructed. My guess is that the kinetic energy is transformed into potential energy. Or maybe it depends on the context of the waves where do the energy goes? Can someone elaborate on that or correct me if I'm wrong? System calls supported in running Kernel Is there a way to get the number of or list of system calls supported by currently running Linux Kernel? So I want to find a way to 'read' the syscall table of a running kernel. Escape velocity from a rotating black hole Under Newton, the escape velocity is  $$v_{esc} = \rm c \ \sqrt{r_s/r}$$  where $\rm r_s=2 \ GM/c^2$. In the nonrotating relativistic case (the Schwarzschild case) the radial escape velocity is the same:  $$v^{\parallel}_{esc} = \rm c \ \sqrt{r_s/r}$$  which is the speed of light, $v={\rm c}$, at the event horizon at $\rm r=r_s$. Since the photon sphere, where an object with $v={\rm c}$ would not escape, but be captured into orbit, is at $r=3 \ {\rm GM/c^2}$, the transverse escape velocity is  $$v^{\perp}_{esc} = \rm c \ \sqrt{r_s/r}\div \sqrt{1-r_s/r}$$  and for the observed (shapiro delayed) velocity ${\rm v}$ the transformation is  $${\rm v}^{\perp}=v^{\perp} \ \sqrt{\rm 1-r_s/r} \ , \ \ {\rm v}^{\parallel}=v^{\parallel} \ (\rm 1-r_s/r)$$  so the observed angular motion is the same as with Newton, yet locally higher, while the observed radial motion is slowed down because of the radial length contraction and time dilation.  But what about the Kerr case where the rotation is significant and the event horizon is at $\rm r=(1+\sqrt{1-a^2}) \ {\rm GM/c^2}$ instead of $\rm r=r_s$?  A testparticle which is locally at rest and therefore corotating with the angular velocity of the frame dragging  $${\Omega = \rm d \phi/ {\rm d t} = {\rm 2 \ a \ r/((a^2+r^2)^2-a^2 (a^2+(r-2) \ r) \sin ^2(\theta ))}}$$  should experience the time dilation factor  $$\rm \dot t = {\rm d t}/ {\rm d}\tau = \sqrt{{\rm ((a^2+(r-2) \ r) (a^2 \cos ^2(\theta )+r^2))/((a^2+r^2)^2-a^2 (a^2+(r-2) \ r) \sin ^2(\theta ))}}$$  relatively to the far away observer, so multiplying $\Omega$ with $\rm \dot t$ should give the local frame dragging angular velocity, which, multiplied with the $\rm \sqrt{x^2+y^2}$ value of the Boyer-Lindquist to cartesian transformation should give the transverse velocity - this is slightly below $\rm c$ for a maximaly rotating black hole (with $\rm a = 1$ - at the edge of the outer equatorial ergosphere where $\rm r=1$ and $\rm R=\sqrt{r^2+a^2}$).  But how do I find the radial and total escape velocity, and what are the rules to convert the shapiro-delayed coordinate velocities $\rm d r/d t \ , \ d \phi/d t \ , d \theta/d t$ to the local 3-velocity components $v^{\parallel}$ and $v^{\perp}$ of a test particle (local meaning relatively to a frame dragged probe with contant $\rm r$)? The radial length contraction seems a bit different than in the Schwarzschild-case, where it is the same as the gravitational time dilation factor... What are some grandiloquent, or simply better, ways of expressing "an idea/thought suddenly came to me", or "an idea/thought struck me", or "I was struck by an idea/thought"? For linear regression, we can check the diagnostic plots (residuals plots, Normal QQ plots, etc) to check if the assumptions of linear regression are violated.  For logistic regression, I am having trouble finding resources that explain how to diagnose the logistic regression model fit. Digging up some course notes for GLM, it simply states that checking the residuals is not helpful for performing diagnosis for a logistic regression fit.  Looking around the internet, there also seems to be various "diagnosis" procedures, such as checking the model deviance and performing chi-squared tests, but other sources state that this is inappropriate, and that you should perform a Hosmer-Lemeshow goodness of fit test. Then I find other sources that state that this test may be highly dependent on the actual groupings and cut-off values (may not be reliable).  So how should one diagnose the logistic regression fit? I have a php file which I will be using as exclusively as an include. Therefore I would like to throw an error instead of executing it when it's accessed directly by typing in the URL instead of being included.  Basically I need to do a check as follows in the php file:  if ( $REQUEST_URL == $URL_OF_CURRENT_PAGE ) die ("Direct access not premitted");   Is there an easy way to do this? Setting Windows PowerShell environment variables One way to prove that there is an infinity of consecutive powerful integers is to just note that there are infinite solutions to the equation $x^2-8y^2=1$, because clearly $x^2$ and $8y^2$ are clearly powerful. But this uses a bit of theory.  I found this problem under an induction worksheet, and all of the surrounding problems are easy. Is there really an elementary solution to the problem?  All of my attempts have been fruitless.  Note: a number is powerful if whenever a prime $p$ divides it we have that $p^2$ also divides it.