Why don't we define $\frac00$ (or other undefined expressions) the same way we define $\sqrt{-1}$? My guess, and my Calculus professor's opinion, is that by doing so we really do not achieve anything. In other words, it's useless to do so. However, how do we know defining undefined expressions is useless? Take for example $\frac00$.
Before, the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had. So we declared $i$ to be a new type of number. How come we don't do the same for other "impossible" equations, such as $x = x + 1$, or $x = 1/0$?  Edit: OK, a lot of people have said that a number $x$ such that $x = x + 1$ would break the rule that $0 \neq 1$. However, let's look at the extension from whole numbers to include negative numbers (yes, I said that I wasn't going to include this) by defining $-1$ to be the number such that $-1 + 1 = 0$. Note that this breaks the "rule" that "if $x \leq y$, then $ax \leq ay$", which was true for all $a, x, y$ before the introduction of negative numbers. So I'm not convinced that "That would break some obvious truth about all numbers" is necessarily an argument against this sort of thing.
What is the reasoning behind limiting "recidivists" to post one question per week? currently restricts recreated accounts to "posting only one question per week..."  I would like to understand why this rate was picked, and not, say, once a day or once a month?  The reason why I ask is that some users limited or banned at Stack Overflow . I expect that learning more about the limitation would help me better understand its impact on Programmers. I am wondering if there is an easy way to inset mesh that is generated from text/curves? When I convert text to mesh it generates a bunch of uneven triangles that don't inset properly. My only solution is slow and tedious and requires manually removing all but the outline edges and refilling with quads. More specifically I am trying to create an engraving of text into a mesh, and rather than a straight bool cut I need the engraved text to inset a bit. The result needs to mesh (not through a shader). Any ideas appreciated. interpreting scatter plots I have plotted my predictors versus my response variable in order to be able to fit a good Poisson model to my data. However i am getting  those scatter plots for nearly all predictors which make me rather worried since including these predictors is making my residual deviance a value of 575 on 571 df(high enough) Thanks If $a^{n}-1$ is prime then $a=2$ and $n$ is prime? The effect of gravitational & Coulomb forces on free electrons in a conductor In a metal, why don’t the free electrons fall to the bottom of the metal due to gravity?  Also, charges in a conductor are supposed to reside on the surface so why don’t the free electrons all go to the surface?   Regarding the first question, my hunch is that it is due to the electric force which is greater than the gravitational force and electric force act along the length of the conductor.  I don't have any intuition for the second. What is least angle regression? Conceptually, I don't understand what least angle regression Least Angle Regression (LARS) is and why it solves LASSO ().  We know that LASSO is:   $$\arg \min_x {\left\| A x - y \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{1}$$  From my understanding LARS is exactly like line search from CGD, where you take the variable that gives you the most desire result and iterate through.  Can someone give a walk through on how LARS solves LASSO? How do I resize partitions using command line without using a GUI on a server? I only have access to the server via a terminal and I can't use graphical tools such as GParted! I want to create a new partition from a part of the root (about 768mb) for swap.  # df -h Filesystem      Size  Used Avail Use% Mounted on /dev/vda         20G  1.3G   18G   7% / udev             10M     0   10M   0% /dev tmpfs           199M  4.9M  194M   3% /run tmpfs           100M   12K  100M   1% /run/user tmpfs           5.0M  4.0K  5.0M   1% /run/lock Cardinality of order types in $\mathbb{N}$ Let be the next equivalence relation on $B= \{R \subseteq \mathbb{N}\times\mathbb{N} \ | \ R \ is \ a \ well-order \ on \  \mathbb{N}\}$ : $$"R \equiv R' \leftrightarrow \ (\mathbb{N},R) \cong (\mathbb{N},R') \ are \ well-ordering \ sets \  isomorphic" $$  Which is the cardinality of $B/\equiv$?  Firstly, we can notice that each well-ordering set is isomorphic to an ordinal $\alpha \ $such that $\alpha &lt; \aleph_{1}$, and so it is enough to count how many ordinals are isomorphic to a subset of $\mathbb{N}$. For example, if we consider $A$ the set of the even numbers and $B$ the set of odd numbers with the usual order relation, $A \oplus B$ is isomorphic to $\omega + \omega$. Another example could be: if we consider the order of $\mathbb{N}$ that we use in Sharkowski Theorem, that order is isomorphic to $ \omega^{2} + \omega$. Working with prime numbers we could build also $\omega^{n} \ \forall n \in \mathbb{N}$, and so $\omega^{\omega}$ and maybe something bigger. However, we know that each countable ordinal is isomorphic to a subset of $\mathbb{Q}$, but I bet that the same result is not true for $\mathbb{N}$ because it is "too small".  What can I do? What are the rules for China's visa-free transit programs? One of my coworkers has provided me with a Bash syntax that I am unfamiliar with.  My Google foo has failed me on figuring out what it does and why/when I should use it.    The command that he sent me was of this form:  someVariable=something command   Initially, I thought that this was equivalent to the following:   someVariable=something ; command   Or  someVariable=something  command   But this doesn't appear to the be case.  Examples:   [Jan-03 11:26][~]$ # Look at the environment variable BAZ. It is currently empty [Jan-03 11:26][~]$ echo $BAZ  [Jan-03 11:27][~]$ # Try running a command of the same format     [Jan-03 11:27][~]$ BAZ=jake echo $BAZ  [Jan-03 11:27][~]$     [Jan-03 11:27][~]$ # Now, echo BAZ again. It is still empty:     [Jan-03 11:27][~]$ echo $BAZ  [Jan-03 11:27][~]$     [Jan-03 11:28][~]$     [Jan-03 11:28][~]$ # If we add a semi-colon to the command, we get dramatically different results: [Jan-03 11:28][~]$ BAZ=jake ; echo $BAZ jake [Jan-03 11:28][~]$ [Jan-03 11:28][~]$ # And we can see that the variable is actually set: [Jan-03 11:29][~]$ echo $BAZ jake [Jan-03 11:29][~]$   What does this syntax do?  What happens to the variable that has been set?  Why does this work? Why can't the Others go around the Wall at sea? As explained in a  earlier , the Wall does not extend all the way to the sea on the East, and the castle there is essentially undefended anyway. Thus, wildlings -- and Stannis' army -- were able to go around it undetected.  We know that the Others and the wights they raise can't cross through the Wall, and I think we can assume that it's built to stand up to any attacks they make trying to bring it down. But there does not appear to be any reason they can't just go around like anyone else.  Possibly related: in the TV Show, we actually see the attempted evacuation of Hardhome, and the Night King allows Jon Snow and his human army and civilians to flee back to their boats to safety, even after raising a massive army of wights. You could interpret this as merely a psychological ploy, but it may be a hint that the dead can't cross the water either.  Has there been any indication (books or show) that the Others/White Walkers and their army of wights are blocked by open water, or are somehow tied to the land of the Frozen North, or that there's any other obstacle keeping them from just sailing/swimming around the Wall? Wrong Java resource bundle loaded Showing 2 Distributions are the Same Let $X_1, X_2, \dots$ be i.i.d. exponentially distributed RVs. For $n = 1,2,\dots$ consider: $Y_n := \max(X_1, X_2, \dots, X_n)$ $U_n := \sum_{i=1}^{n}\frac{X_i}{i}$ Show that $Y_n$ and $U_n$ have the same distribution   What I've tried: $P(Y_n&lt;y)= P(X_i&lt;y)^n = (1- e^{\lambda y})^n =&gt; P(Y_n=y) = n(1- e^{\lambda y})^{n-1}$ But I get stuck with $U_n$. I tried an MGF, $U_n$ evaluates nicely but then $Y_n$ gets messy. Any thoughts? Determine which app is drawing on top of other apps? Is there a way to determine which application is drawing on top of other applications?  For example, Facebook Messenger app can draw the chat bubbles on top of everything, but how do I know which app is doing this?  My problem:  I have an app which is drawing ads on top of everything and I wanna know which app is doing this. Not just to uninstall it, but to warn everyone that uses this app. (I didn't install any application recently and I already verified the recent updates.)  I tried to use the Dump View Hierarchy for UI Automator from the Android Studio, but it seems that the applications that draw on top can't be selected this way.  (Click image to enlarge) When I dock my Windows 7 laptop, I want it to prefer the wired ethernet connection over WiFi.  This is a pretty straightforward thing to do on my Mac - I just reorder my network preferences, and it "does the right thing." I just can't figure out how to achieve the same thing on my Win7 laptop.  So, when I'm docked, it connects to WiFi, and then fails to connect to servers on the local wired network. How do I fix this? How can I force a long string without any blank to be wrapped? I have a long string (a DNA sequence). It does not contain any whitespace character.  For example:  ACTGATCGAGCTGAAGCGCAGTGCGATGCTTCGATGATGCTGACGATGCTACGATGCGAGCATCTACGATCAGTCGATGTAGCTAGTAGCATGTAGTGA   What would be the CSS selector to force this text to be wrapped in a html:textarea or in a xul:textbox? According to some source or other (I forget which now) it is theoretically impossible for bumblebees to fly by virtue of their size/bulk/aerodynamic properties. Is this old adage apocraphyal or true? And if its true, how come they do fly? First order differential equations with 2 varriables Does anybody know how to solve the following differntial equation?  $$w^2\cdot r+\frac{dw}{dt}\cdot r+\frac{dr}{dt}\cdot w=3g$$  where w and r are a function of time so $w(t)$ and $r(t)$. $g$ is a constant  EDIT: This equation came from a thought problem of mine which I will now explain since I still do not know how to approach the problem.  Basically I want to design a spiral track in which a vehicle will accelerate. However the total acceleration needs to be constant.   The vehicle will have 2 acceleration components, a radial component ($a_r$) and a change in speed component ($a_s$). Since we want constant acceleration: $a_r + a_s = constant$  From circular motion we know that $a_r = \omega ^2 r$ where $r$ is the radius, and $\omega$    Further we know that $v=\omega*r$, therefore: $a_s=\frac{d}{dt}(\omega r)$  For constant acceleration $a_{total} = a_r + a_s = \omega ^2 r + \frac{d}{dt}(\omega r) =constant$  Further usefull information: Length of the track covered at a certain time $l=v*t=\omega r t$  With all this the radius will be a function of time, $r(t)$ and the angular velocity will be a function of time, $\omega (t)$  Boundary conditions: $r(0)  = 0$ $\omega (0) = 0$  Now I would like to find a possible set of solutions to create a track by implementing end boundary conditions that i choose e.g. $v(t_{end})=100 [m/s]$ and $l(t_{end})=5000 [m]$  Here is a diagram for clarity: How to I make Cisco WebEx work with 13.10 64bit? I'm having a very hard time getting webex to work under Saucy.  Up until now I've been able to just install a java plugin, install ia32-libs, and I was good to go.  With Saucy ia32-libs is gone and it's up to us to figure out which 32-bit libraries we need to install.    So the question is, how do I install WebEx on 64bit Saucy without ia32-libs? What happens if both players have the ability to pump their creature an unlimited number of times? Say that both players each had 2 , 2 , and a . (Or any other way to get unlimited mana, and a Nantuko Shade).  Player 1 attacks with his , and player 2 blocks with his . Before damage is assigned, each has chances to pump up their  as much as they want. They can't just declare them to be "infinite," but each can always one-up the other player. What happens?  It seems there would be a loop of events, but would one player eventually be forced to break the loop as per rule 714.3? Is it considered the same game state after each player pumps their creature a bunch as it was before?  This was discussed at length on the Draw3Cards Magic Q&amp;A site (which is no longer available), but no conclusion was reached. Some felt that the game would simply be a draw, others felt that the active player would be forced to make a different decision to break the loop. Others felt that you would just have to call a judge and see what he decides.