Adventurer enters an other-dimensional realm where he repeatedly dies unexpectedly and wakes up in reality
Pre-1972 sci-fi short story or novel: alien(?) tunnel where people try new moves and get destroyed if they're not the correct ones
What's the point of Hamiltonian mechanics? I've just finished a Classical Mechanics course, and looking back on it some things are not quite clear. In the first half we covered the Lagrangian formalism, which I thought was pretty cool. I specially appreciated the freedom you have when choosing coordinates, and the fact that you can basically ignore constraint forces. Of course, most simple situations you can solve using good old $F=ma$, but for more complicated stuff the whole formalism comes in pretty handy.  Then in the second half we switched to Hamiltonian mechanics, and that's where I began to lose sight of why we were doing things the way we were. I don't have any problem understanding the Hamiltonian, or Hamilton's equations, or the Hamilton-Jacobi equation, or what have you. My issue is that I don't understand why would someone bother developing all this to do the same things you did before but in a different way. In fact, in most cases you need to start with a Lagrangian and get the momenta from $p = \frac{\partial L}{\partial \dot{q}}$, and the Hamiltonian from $H = \sum \dot{q_i}p_i - L$. But if you already have the Lagrangian, why not just solve the Euler-Lagrange equations?  I guess maybe there are interesting uses of the Hamiltion formalism and we just didn't do a whole lot of examples (it was the harmonic oscillator the whole way, pretty much). I've also heard that it allows a somewhat smooth transition into quantum mechanics. We did work out a way to get Schrödinger's equation doing stuff with the action. But still something's not clicking.  My questions are the following: Why do people use the Hamiltonian formalism? Is it better for theoretical work? Are there problems that are more easily solved using Hamilton's mechanics instead of Lagrange's? What are some examples of that? I just graduated, and I'm starting a new job soon. My employer has pretty good policy for matching 401(k) contributions, so I want to invest as much as possible, especially during this first year.  My question is: should I invest in a traditional 401(k), a Roth 401(k), or some combination of the two? On the one hand, tax rates may be much higher when I retire, and I might be in a higher tax bracket, so a Roth 401(k) would be good. On the other hand, I might be in a lower tax bracket when I retire, so a traditional 401(k) might be better. It's hard to know where I will stand in 30 years.  Is this the same trade-off as with a traditional / Roth IRA, or is there any other tax difference I should know about? Can I be admitted to graduate school in a different field from my degree?  Specific cases include:   If I've taken plenty of advanced courses in field X in the process of completing a degree in another field, can I apply to graduate school in X? What if I haven't taken many courses in X, but I have acquired a good grasp of X through self-study or working in a related field? What if I've never studied X, but I have done very well in an unrelated field?  Could I be admitted to graduate school in X on the basis of general intellectual promise, and then make up the missing background after enrollment?   Note that this question is an attempt to provide a comprehensive answer, to avoid the need for a profusion of field-specific questions on this topic (see the associated ). Please feel free to edit the question or answer to improve them. How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$? If statement evaluating true no matter what. Python 3.3.0 Deleting a channel on slack.com Suppose that $ f $ is entire and that for each $ z $, either $ |f(z)| \leq 1 $ or $ |f^\prime (z) |\leq 1 $. Prove that $ f $ is a linear polynomial. My question is in the title. I'm a little lost in how to solve this problem. There is a hint associated with the problem that states the following:   Use a line integral to show that $ |f(z)| \leq A + |z| $ where A = $\max\{1, |f(0)|\}$  Thanks for the help. Trigonometry Olympiad problem: Evaluate $1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$ Find the value of   $$1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$$   My attempt  I converted the $\sin$ functions which have arguments greater than $90^\circ$ to $\cos$ but I have gone no where with it!  I also tried using double angle formula for the angles which are even. pygame.event.get() not returning any events when inside a thread Windows 7 SP1 Windows Update stuck checking for updates I installed Windows 7 fresh and installed SP1. Now, when I try to check manually for Windows Updates it just hangs on the Checking for updates screen.    I tried running the tools in , but this did not fix the issue either:  No matter what I do it just hangs on the "Checking for updates..." screen and goes no further. Bash/Korn shell script edited on Windows throws error '...^M: not found' Planet where humans fight for access to algae Android: fragments overlapping issue "Reality" of EM waves vs. wavefunction of individual photons - why not treat the wave function as equally "Real"? In thinking how to ask this question (somewhat) succinctly, I keep coming back to a Microwave Oven.  A Microwave Oven has a grid of holes over the window specifically designed to be smaller in diameter than the wavelength of the microwaves it produces, yet larger than the wavelengths of the visible light spectrum - this is so you can watch your food being heated without getting an eye full of microwaves.  The "realness" of electromagnetic waves seems indisputable - both from the microwave example above, and also because if I want to broadcast a radio wave with a certain wavelength, then I need to make sure I have an antenna of corresponding length to produce the wave I'm looking for.  Furthermore, we discuss and treat these waves as real, measurable "objects" that exist and can be manipulated.     Now, if I want to describe the behavior of my Microwave Oven in the framework of QM (let's pretend my oven is going to only produce 1 photon of energy corresponding to a wavelength of the microwave spectrum for simplicity) then I'll describe the behavior of that photon as a wavefunction that evolves over time and gives a probability distribution within my microwave that similarly does not allow the photon to pass through the safety grid and exit the oven cavity giving me a retinal burn.  The difference is, the wavefunction is never treated as something "real" in this description.  When the safety grid is described as working to protect you because it has holes smaller than the wavelength of the classical waves it's blocking, this is a useful description that seems to describe "real" objects/be-ables.  While it is possible to describe why an individual photon has low probability of passing through the same grid but extended physical properties such as wavelength (in space) are treated as non-real because we're dealing with a point particle and with behavior described by something we also treat as non-real (the wavefunction); it seems unclear to me why we insist this wave function which predicts behavior of physical measurements so well is somehow "non-real."    Put another way, if we have no problem treating EM waves as "Real," then why do we insist on treating the wavefunction that describes the same behavior as "unreal?"  I understand there is recent research (Eric Cavalcanti and his group for one) trying to argue this point, but as every respectable physics professor, I've ever encountered, has treated the wavefunction as an indisputably non-real mathematical tool, I needed to ask this community for an answer. How do I change the full background color of the console window in C#? If $\left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right)$ If $\displaystyle \left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right) = $  $\bf{My\; Try::}$ Given $\displaystyle \left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\Rightarrow 1+\sin \phi\cdot \cos \phi+\sin \phi+\cos \phi = \frac{5}{4}.$  So $\displaystyle \sin \phi+\cos \phi+\sin \phi \cdot \cos \phi = \frac{1}{4}\Rightarrow \left(\sin \phi+\cos \phi\right) = \frac{1}{4}-\sin \phi \cdot \cos \phi.$  Now $\displaystyle \left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right) = 1-\left(\sin \phi+\cos \phi\right)+\sin \phi \cdot \cos \phi =\frac{3}{4}+\sin 2\phi$  Now How can I calculate $\sin 2\phi.$  Help me, Thanks How do I filter out addresses using HTTPS for a D-Link router? I recently got a new router, a DRL-600L Router from D-Link, and was setting up some filters to block out certain websites.  One problem I am having is blocking out addresses that use https instead of http.  For example, I'm trying to block out Facebook, because Facebook.   So far, the filter filters http://www.facebook.com perfectly fine, but for the url https://www.facebook.com, it won't block that.  How can I block those kinds of websites? I have a very large dataset and about 5% random values are missing. These variables are correlated with each other. The following example R dataset is just a toy example with dummy correlated data.  set.seed(123)  # matrix of X variable  xmat &lt;- matrix(sample(-1:1, 2000000, replace = TRUE), ncol = 10000) colnames(xmat) &lt;- paste ("M", 1:10000, sep ="") rownames(xmat) &lt;- paste("sample", 1:200, sep = "") #M variables are correlated   N &lt;- 2000000*0.05 # 5% random missing values  inds &lt;- round ( runif(N, 1, length(xmat)) ) xmat[inds] &lt;- NA  &gt; xmat[1:10,1:10]          M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 sample1  -1 -1  1 NA  0 -1  1 -1  0  -1 sample2   1  1 -1  1  0  0  1 -1 -1   1 sample3   0  0  1 -1 -1 -1  0 -1 -1  -1 sample4   1  0  0 -1 -1  1  1  0  1   1 sample5  NA  0  0 -1 -1  1  0 NA  1  NA sample6  -1  1  0  1  1  0  1  1 -1  -1 sample7  NA  0  1 -1  0  1 -1  0  1  NA sample8   1 -1 -1  1  0 -1 -1  1 -1   0 sample9   0 -1  0 -1  1 -1  1 NA  0   1 sample10  0 -1  1  0  1  0  0  1 NA   0   Is there a (best) way to impute missing values in this situation? Is the Random Forest algorithm helpful? Any working solution in R would be much appreciated.  Edits:  (1) Missing values are randomly distributed among the variables and samples.As number of variables is very large (here in the example - 10000), while the number of samples is small here in the above dummy example it is about 200. So when we look at any sample over all the variables (10000), there is high chances that there will be missing value at some variable - due to large number of variables. So just deleting the sample is not option.   (2) The variable can be treated both as quantitative or qualitative (binary) in the process of imputing. The only judgement is how well we can predict it (accuracy). So predictions like 0.98 instead of 1 might be acceptable rather 0 vs 1 or -1 vs 1. I might need to tradeoff between computing time and accuracy.   (3)  The issue I have thinking how overfitting can affect the results as the number of variables are large compared to number of samples.   (4) As the total quantity of missing values is about 5% and is random (not concentrated in any variables or samples as precaution was taken to remove the variables or samples that have very high missing values)   (5) Making data complete for analysis is first objective and accuracy is secondary. So not too sensitive to accuracy. What is the difference between range and xrange functions in Python 2.X? How do I remove the Unity launcher? How can we remove the launcher that come with Unity on this new Ubuntu version ?