How do I solve for x? $y=e^x+x+1$
What is the inverse function of $e^x +x$?
Will my device work with Ubuntu? Why can we use sample variance of the treatment group to estimate population variance of the controls? 1.  Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an explanation that I need some help with: that the de Broglie wavelength and the wavelength of an elastic wave do not show similar properties under a Galilean transformation. He basically says that both are equivalent under a gauge transform and also, separately by Lorentz transforms. This, accompanied with the observation that $\psi$ is not observable, so there is no "reason for it being real". Can someone give me an intuitive prelude by what is a gauge transform and why does it give the same result as a Lorentz tranformation in a non-relativistic setting? And eventually how in this "grand scheme" the complex nature of the wave function becomes evident.. in a way that a dummy like me can understand.  2.   A wavefunction can be thought of as a scalar field (has a scalar value in every point ($r,t$) given by $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{C}$ and also as a ray in Hilbert space (a vector). How are these two perspectives the same (this is possibly something elementary that I am missing out, or getting confused by definitions and terminology, if that is the case I am desperate for help ;)   3.   One way I have thought about the above question is that the wave function can be equivalently written in $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{R}^2 $ i.e, Since a wave function is complex, the Schroedinger equation could in principle be written equivalently as coupled differential equations in two real functions which staisfy the Cauchy-Riemann conditions. ie, if $$\psi(x,t) = u(x,t) + i v(x,t)$$ and $u_x=v_t$ ; $u_t = -v_x$ and we get $$\hbar \partial_t u  = -\frac{\hbar^2}{2m} \partial_x^2v + V v$$ $$\hbar \partial_t v  = \frac{\hbar^2}{2m} \partial_x^2u - V u$$ (..in 1-D) If this is correct what are the interpretations of the $u,v$.. and why isn't it useful. (I am assuming that physical problems always have an analytic $\psi(r,t)$). Moderators currently have a tool to add a Post Notice that says:     insufficient explanation       We're looking for long answers that provide some explanation and   context. Don't just give a one-line answer; explain why your answer is   right, ideally with citations. Answers that don't include explanations   may be removed.   However, there seems to be no equivalent wording for one-line questions.  Could the wording be changed to something like below to cover both one line questions and one-line answers?     insufficient explanation       We're looking for longer posts that provide some explanation and context.          Don't just ask a one-line question; provide some background like a brief description of your research prior to asking   Don't just give a one-line answer; explain why your answer is   right, ideally with citations.          Questions and answers that don't include explanations may be removed.   Sometimes such questions can simply be closed for being unclear or too broad but on the two sites I moderate there is an aversion to closing questions from new users, so to give them some guidance about what is expected, without spending time writing the same comment over and over, a post notice like this would provide a good "couple of clicks" alternative. An inequality from Littlewood's Miscellany Define a linear operator $T:C[0,1] \to C[0,1]$ as follows: $$Tf(x) = f(x) + \int_0^x f(u)du$$ It is easy to show that $T$ is a bounded linear operator. The statement also (1) claims that $T$ is one-to-one and onto, and (2) asks to compute $T^{-1}$. (Hence $T$ is an isomorphism due to a corollary of Open Mapping Theorem)  I need help to prove $T$ is onto: given any $g \in C[0,1]$, how to decompose it into a sum between a continuous $f$ and its antiderivative. I have trouble decomposing $g(x)=|x|$, not to mention bizarre functions like Weierstrass function. It seems impossible.  Thanks in advance for help! Does the sequence $$\displaystyle \frac{n}{\sum\limits_{k=1}^{n}\Big(\frac{k}{k+1}\Big)^k}$$   converge?   Attempt. Since $\Big(\frac{k}{k+1}\Big)^k \rightarrow 1/e\neq 0$ and the terms are positive, the series $\sum\limits_{k=1}^{\infty}\Big(\frac{k}{k+1}\Big)^k$ diverges to $+\infty$. I find hard to determine if $n$ or the sum goes faster to $+\infty.$   Thanks in advance. Shocking? - MacBook stubby earthed charger UK I have no idea why. But apple have not earthed the stubbly plug for the macbook pro transformer. So on the short lead the only way the charger is earthed is when I touch it or the macs unibody which gives you a sort of vibrating feeling on your palms.  When I went to the genius bar to complain about it they said that it was completely 'normal'. Where their definition of normal is that every MacBook does it. I have had 4 MacBooks and all have done it.    The genius did say that he though he has seen a few earthed 3pin plugs. So my question is where can I pick one up? or how could we force apple to earth the stubby plugs and recall them? Show that no application $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, of $C^k$ class, $k \geq 1$ can be injective How can I proof this:     Show that no application $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, of $C^k$ class, $k \geq 1$ can be injective, i.e., there are $A,B \in \mathbb{R}^2$ such that $A \neq B$ and $f(A) = f(B)$ (Hint: show that given $(a,b) \in \mathbb{R}^2$ in every open neighbourhood of $(a,b)$ there are points $(c,d),(c_1,d_1)$ where ${\partial}_1f(c,d) \neq 0$ or ${\partial}_1f(c_1,d_1) \neq 0$).   The problem is that I have no idea how to use the hint.  Thanks for all your comments. "E:Read error - read (5: Input/output error), E:The package lists or status file could not be parsed or opened." So I keep trying to install things on my laptop and whenever I try to use sudo apt-get I get the same errors every  time,  SystemError: E:Read error - read (5: Input/output error), E:The package lists or status file could not be parsed or opened.   Also whenever I try to open software Center through my desktop Icon it will just freeze leaving me to force quit.  If I try to use the command software-center in my terminal I get the longest list in the world which is,   che.open() Traceback (most recent call last):   File "/usr/lib/python2.7/dist-packages/gi/overrides/GLib.py", line 629, in &lt;lambda&gt;     return (lambda data: callback(*data), user_data)   File "/usr/share/software-center/softwarecenter/db/pkginfo_impl/aptcache.py", line 261, in open     self._cache = apt.Cache(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 105, in __init__     self.open(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 150, in open     self._cache = apt_pkg.Cache(progress) SystemError: E:Read error - read (5: Input/output error), E:The package lists or status file could not be parsed or opened. 2013-10-13 10:13:52,685 - softwarecenter.db.pkginfo_impl.aptcache - INFO - aptcache.open() Traceback (most recent call last):   File "/usr/lib/python2.7/dist-packages/gi/overrides/GLib.py", line 629, in &lt;lambda&gt;     return (lambda data: callback(*data), user_data)   File "/usr/share/software-center/softwarecenter/db/pkginfo_impl/aptcache.py", line 261, in open     self._cache = apt.Cache(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 105, in __init__     self.open(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 150, in open     self._cache = apt_pkg.Cache(progress) SystemError: E:Read error - read (5: Input/output error), E:The package lists or status file could not be parsed or opened. 2013-10-13 10:13:57,365 - softwarecenter.db.pkginfo_impl.aptcache - INFO - aptcache.open() Traceback (most recent call last):   File "/usr/lib/python2.7/dist-packages/gi/overrides/GLib.py", line 629, in &lt;lambda&gt;     return (lambda data: callback(*data), user_data)   File "/usr/share/software-center/softwarecenter/db/pkginfo_impl/aptcache.py", line 261, in open     self._cache = apt.Cache(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 105, in __init__     self.open(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 150, in open     self._cache = apt_pkg.Cache(progress) SystemError: E:Read error - read (5: Input/output error), E:The package lists or status file could not be parsed or opened.   That's only about one-tenth of the list which is too long to read. I need help, if anyone has an answer please respond Suppose $ T \subset \mathbb{C} $. Show that the corresponding set $ S \subset \Sigma $ is  a. a circle if $ T $ is a circle. b. a circle minus (0, 0, 1) if $ T $ is a line.  Here we are defining $ \Sigma $ to be the Riemann sphere, given by the set: $$ \Sigma = \left \{(\xi, \eta, \zeta) : \xi^{2} + \eta^{2} + (\zeta - \frac{1}{2})^{2} = \frac{1}{4} \right \} $$  To take a point from $ \mathbb{C} $ to $ \Sigma $ we can use the following:  $$ \xi = \frac{x}{x^{2} + y^{2} + 1}; \eta = \frac{y}{x^{2} + y^{2} + 1}; \zeta = \frac{x^{2} + y^{2}}{x^{2} + y^{2} + 1} $$  We define a circle on $ \Sigma $ to be the intersection of a plane of the form $ A\xi + B\eta + C\zeta = D $ with $ \Sigma $. We also know the converse of this problem is true, that the intersection above yeilds a set in $ \mathbb{C} $ with the following property:  $ (C - D)(x^{2} + y^{2}) + Ax + By = D $. As you can see, when C = D, then an equation for a line is yeilded, otherwise it is a circle.   I really am at a  loss about how to solve this problem. The only thing I can think to do is to pick 3 points on a circle or radius $ r $ with center $ z_{0} $, use these points to find two vectors in $ \Sigma $, take their cross product to get a normal vector, use this normal vector to get a plane. Once I have the plane in form $ A\xi + B\eta + C\zeta = D $ then I could prove that the circle I had chosen corresponds exactly with $ (C - D)(x^{2} + y^{2}) + Ax + By = D $. Is there not an easier, less computation way to do this? VirtualBox - how to increase video memory? "I wish I didn't do" vs. "I wish I hadn't done" This question has been answered before in many places but I just need some confirmation with my specific setup. I have Linux Mint installed and I just want a fresh Ubuntu 12.04 install to completely replace Mint. It's dual booted with Windows 7 and I want that to remain completely in tact.  Anyway, I'm currently running the Live CD installer. I've said I want to specify my own partitions and I'm at the following screen:     The list at the bottom expanded:     I confused by all the different partitions. Seems like I should just check the box beside sda5. But what about sda6? Do I just leave that alone?  Also just for confirmation, is the "right" way to replace Mint, or should I be deleting it first? Set "older" kernel as default grub entry I installed a mainline kernel for testing purposes. I would like to set grub to boot from the older kernel by default.  I know I can set the GRUB_DEFAULT=0 setting for the first page of grub but how do I set it to boot by default from one kernel in the second page (Advanced page)?  I would prefer doing this without installing other software (ex. grub-customizer).     Related Questions: I would like to transfer files between two remote hosts using on local shell, but it seems rsync doesn't support synchronisation if two remotes are specified as follow:  $ rsync -vuar host1:/var/www host2:/var/www The source and destination cannot both be remote.   What other workarounds/commands I could use to achieve similar results? Outcast boy meets girl underwater Here's another one I can't find, but somebody will probably recognize right away. A young man has been thrown out of respectable society. To make sure he stays away from people (and they stay away from him), those who have condemned him alter his body so that he always smells bad. Of course, no one can smell underwater - and that's where he meets a girl, while they're both swimming underwater in a swimming pool. I don't remember a lot of details about this story - but I remember the main character was not very nice - he was often mean to other people. Whether he'd always been like that, or whether his nastiness was because everyone had rejected him, I can't remember. I would have read this story in the mid-seventies, but it was probably written earlier. Uncertainty principle and multiple observers Exists a dedicated non-O(n²) Algorithm for avoiding to have overlapping elements? I am generating positions in a random fashion.  In the game world, instantiated elements have a radius, and shouldn't overlap.  The problem is that they will overlap because I use a usual random number generator combined with other fanciness. I can't use other random number generation algorithms where I can guarantee that the elements will never overlap.  A way to solve this is to use a spatial data structure and search for the nearest neighbors in that tree. When I insert an element into that tree, I update it accordingly.  This solution would do the work and it would be fast but my question is:  Is there a dedicated algorithm without that Tree implementation complexity which is faster than O(n²)? Motivation behind the definition of ideal class group