Different definitions for exterior derivative? Let $\omega \in \Omega^k(M)$ and $X_i \in \Gamma(TM)$. In Spivak Volume I page 213 the exterior derivative is given invariantly by the formula \begin{align*}d \omega(X_0, \ldots, X_k) = &amp; \sum_i (-1)^i X_i . (\omega(X_0, \ldots, \hat{X_i}, \ldots, X_k)) \\ &amp;+ \sum_{i&lt;j} (-1)^{i+j} \omega([X_i, X_j], X_0, \ldots, \hat{X_i}, \ldots, \hat{X_j}, \ldots, X_k) \\ \end{align*} whereas in Kobayashi and Nomizu Volume I page 26 the exterior derivative is given by almost the same formula, but with a factor of $\frac{1}{k+1}$ in front of the sums: \begin{align*}d \omega(X_0, \ldots, X_k) = &amp; \frac{1}{k+1}\sum_i (-1)^i X_i . (\omega(X_0, \ldots, \hat{X_i}, \ldots, X_k)) \\ &amp; + \frac{1}{k+1} \sum_{i&lt;j} (-1)^{i+j} \omega([X_i, X_j], X_0, \ldots, \hat{X_i}, \ldots, \hat{X_j}, \ldots, X_k) \\ \end{align*} I'm confused about where this extra factor is coming from. Even if Spivak and Kobayashi and Nomizu use different conventions for the wedge product, shouldn't the invariant formula for the wedge product be the same?
different possible definitions of the exterior derivative? In Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces"  we have the formula for the exterior derivative  $(p+1)d\omega(X_{1}, \ldots X_{p+1})=\Sigma_{i=1}^{p+1} (-1)^{i+1} X_{i} \cdot \omega(X_{1}, \ldots , \hat{X_{i}}, \ldots X_{p+1})+\Sigma_{i&lt;j} (-1)^{i+j}\omega([X_{i}, X_{j}], X_{1}, \ldots , \hat{X_{i}}, \ldots \hat{X_{j}}, \ldots , X_{p+1})$  but in other sources on-line they have the same formula without the factor of $p+1$ on the left-hand side.  Are there two different definitions of the exterior derivative, or is there an error in Helgason?
Softmax vs Sigmoid function in Logistic classifier? What decides the choice of function ( Softmax vs Sigmoid ) in a Logistic classifier ?  Suppose there are 4 output classes . Each of the above function gives the probabilities of each class being the correct output . So which one to take for a classifier ? Dwellers return automatically after finding a low amount of items in Wasteland After running an upgrade from Ubuntu 11.10 (Oneiric Ocelot) to 12.04LTS (Precise Pangolin), a window popped up saying that the upgrade failed and that I should run sudo dpkg --configure -a.  Running that command gave me:  sudo dpkg --configure -a dpkg: error: parsing file '/var/lib/dpkg/status' near line 55497 package 'indicator-displex':  blank line in value of field 'Description   Trying to run apt-get install -f gives me many dependency errors along with that same dpkg error. How to find out if an item is present in a std::vector? What kind of lubrication should I use for a lock? I recently made the Yale lock on my front door much easier to open by spraying some WD-40 in there.  However, I just saw  on WD-40 not being a “true” lubricant, and wondered whether it was appropriate to use on Yale locks? $X$ is homeomorphic to $X\times X$ (TIFR GS $2014$) Question is :  Suppose $X$ is a topological space of infinite cardinality which is homeomorphic to $X\times X$. Then which of the following is true:   $X$ is not connected. $X$ is not compact $X$ is not homeomorphic to a subset of $\mathbb{R}$ None of the above.   I guess first two options are false.  We do have possibility that product of two connected spaces is connected.  So, $X\times X$ is connected if $X$ is connected. So I guess there is no problem.  We do have possibility that product of two compact spaces is compact.  So, $X\times X$ is compact if $X$ is compact. So I guess there is no problem.  I understand that this is not the proof to exclude first two options but I guess the chance is more for them to be false.  So, only thing I have problem with is third option.  I could do nothing for that third option..  I would be thankful if some one can help me out to clear this.  Thank you :) Let $f$ be a nonconstant element of ring $\mathbb Z[x]$. Prove that $\langle f \rangle$ is not maximal in $\mathbb Z[x]$.   Let us assume $\langle f \rangle$ is maximal. Then $\mathbb Z[x] / \langle f \rangle$ would be a field. Let $a \in \mathbb{Z}$. Then $a + \langle f \rangle$ is a nonzero element of this field, hence a unit. Let $g + \langle f \rangle$ be its inverse. Then $a g - 1 \in \langle f \rangle$, hence $ag(x)-1 = f(x)h(x)$ for some $h \in Z[x]$, hence $ag(0) + f(0)h(0) = 1$, thus $(a,f(0))=1$ for all $a \in \Bbb Z$, contradiction, hence the proof.  Is my argument correct? Is there any other method? I have a PC with an OEM install of Win 8 updated to 8.1.  It has 3 physical hard drives in it.  The drive where Windows is installed has a LOT of partitions on it already.    Can I install Ubuntu on a different physical hard drive than Windows, and be able to dual-boot?  I am able to boot to a live USB of Ubuntu 14.04, so I think I'm about ready to proceed.  But if I can make my installation easier by using a different physical drive, I'd like to do that. What happened after the Ring was dropped into Mount Doom? How do I access the index in a for loop like the following?  ints = [8, 23, 45, 12, 78] for i in ints:     print('item #{} = {}'.format(???, i))   I want to get this output:  item #1 = 8 item #2 = 23 item #3 = 45 item #4 = 12 item #5 = 78   When I loop through it using a for loop, how do I access the loop index, from 1 to 5 in this case? Should I ask questions I already know the answer to and answer them myself? Violation of linearity assumption in Logistic Regression I built a logistic model with 7 continuous independent variable and 8 categorical (dummy) variables in R. I tested for the linearity assumption using the Box Tidwell test and all my variables show high significance.  I am wondering now, if the assumption is violated what it would mean for the coefficient estimates and if I still can run the model with caution.   I read about the approach of using regression splines but I am confused how I would determine those and interpret the model then. Permanently removing apache2 It seems a simple apt-get remove apache2 does not completely remove apache2 as I can still see it on one of the processes when running top. How does one remove apache2 completely on his ubuntu server?  It's not removed indeed:  ~# which apache2 /usr/sbin/apache2 ~# whereis apache2 apache2: /usr/sbin/apache2 /etc/apache2 /usr/lib/apache2 /usr/share/apache2 /usr/share/man/man8/apache2.8.gz   But when I do apt-get remove apache2 again:  # apt-get remove apache2 Reading package lists... Done Building dependency tree Reading state information... Done Package apache2 is not installed, so not removed 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Inner product of two continuous maps is continuous Let f,g be continuous maps of S in $\mathbb{R}^n$, to $\mathbb{R}^m$. Show the inner product h(x) = $\langle f(x), g(x)\rangle$ is continuous.  My attempt: Since f,g are continuous, their product is a continuous mapping, so for $\epsilon \gt 0\: t.e\:  \delta \gt 0$ such that $\parallel x - a\parallel \lt \delta $ implies $\parallel \langle f(x),g(x)\rangle - \langle f(a)g(a)\rangle \parallel \le\parallel f(x)g(x) - f(a)g(a)\parallel \lt \epsilon$ by schwartz inequality. Group with order $p^2$ must be abelian . How to prove that? The exchange of photons gives rise to the electromagnetic force Pardon me for my stubborn classical/semiclassical brain. But I bet I am not the only one finding such description confusing.  If EM force is caused by the exchange of photons, does that mean only when there are photons exchanged shall there be a force? To my knowledge, once charged particles are placed, the electromagnetic force is always there, uninterruptedly. According to such logic, there has to be a stream of infinite photons to build EM force, and there has to be no interval between one "exchange event" to another. A free light source from an EM field? The scenario is really hard to imagine.  For nuclei the scenario becomes even odder. The strong interaction between protons is caused by the exchange of massive pions. It sounds like the protons toss a stream of balls to one another to build an attractive force - and the balls should come from nothing.  Please correct me if I am wrong: the excitations of photons and pions all come from nothing. So there should be EM force and strong force everywhere, no matter what type of particles out there. Say, even electrical neutral, dipole-free particles can build EM force in-between. And I find no reason such exchanges of particles cannot happen in vacuum.   Hope there will be some decent firmware to refresh my classical brain with newer field language codes. I can join the server and sometimes the map would load for the server but i would just fall through the map and it would dissconect me and say timed out, any solutions? Short story identification: society partitioned off by high wall Reading another question today reminded me of a short story I read in the early 90s (probably written in 60s or 70s though). A society is blocked off by a really high wall; the story follows a girl who tries to get over the wall. I think she manages it with a hot air balloon. When she reaches the other side she meets an older man who had managed to scale the wall in his youth - his solution had been to use a flock of birds to carry him.  UPDATE  I seem to recall that it splits off an elite group of people - getting over the wall is like a rite of passage. Most people don't even bother to get over it (or even be concerned by its existence at all), but a few try. There might be some magic involved (on the other / better side), or they might just be more advanced technologically.  FINAL UPDATE  As pointed out in the answer, and confirmed by me re-reading the story, I was mistaken in a few details. The main protagonist is not a girl, it's a boy called Porgie. He doesn't use a balloon, it's a glider he made, assisted with a broom. Magic is involved, but it's on the inside, with technology on the outside. Why with two classes, LDA gives only one dimension? Calculations in quotient rings