Translation of travelling wave is again a travelling wave, thus $0$ is an eigenvalue: Why?
Linearization of PDE: $0$ is an eigenvalue since all translates of travelling waves are also travelling waves
I've come across a paper that mentions the fact that matrices commute if and only if they share a common basis of eigenvectors. Where can I find a proof of this statement? Is  && statement() the same as if() statement()? Add a "back to top" link The left-nav is great - I like it, users who don't can disable it. Perfect world. Now it's stickied it's always there too, which is really useful.  When I'm navigating in areas not directly linked to in the left nav and have scrolled down, I often want to re-use the other navigation. This means scrolling up. Which is fine, of course it's how websites work.  However, if the left-nav had a "Top" link which took the user to the very top of the page it would be great for getting back to specific contextual menus and things that are not linked to in the left-nav.  For example, if I'm viewing questions on a specific tag and I've scrolled down, I sometimes want to get back up to the top to remove a tag, or change from "Latest" to "Featured" etc. Although clicking the "Questions" link in the left-nav takes me back to the top, it removes the tag(s) I was searching with.  There are other use cases where this would be useful, like long questions and answers (Etc). So I think a "Top" link in the left nav would be very useful.    I think this is different to the  as it doesn't mention the left-nav, and my question is suggesting this idea because we have a good place to put it and it'll be a useful addition to the left-nav. Does adding two linear equations will result in a line which will pass through an intersection of the linear equations? I was wondering why it is almost impossible to find a geometrical explanation of why adding two linear equations helps us to find a solution of a system of linear equations? Am I right that adding two linear equations will result in an equation of a line which will pass through a point where two linear equations intersect?  If it is right then I completely don't understand why such a crucial point in understanding how to solve systems of equations is never properly taught even at a university level. I need to calculate the following indefinite integral:  $$I=\int \frac{1}{\cos^3(x)}dx$$  I know what the result is (from Mathematica):  $$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$  but I don't know how to integrate it myself. I have been trying some substitutions to no avail.   Equivalently, I need to know how to compute:  $$I=\int \sqrt{1+z^2}dz$$  which follows after making the change of variables $z=\tan x$. Weighting features prior to SVM What are the highest DC and AC voltages or currents that can be considered as safe? Summation by parts (finite calculus) of $\sum\limits_{i=0}^{n}i\cdot a^i$ I'm studying the calculus of finite differences and have read about summation by parts:  $\sum f(x)\Delta g(x) = f(x)g(x) - \sum g(x+h)\Delta f(x)$.  The tutorials I'm using go a bit woolly at the point of introducing this technique: they use examples where they set $\Delta g(x)$ to $2^x$, and don't really show how to use it for anything a bit harder. I'm at a loss for basic examples.  I wondered if it is possible to determine  $S.n = \sum\limits_{i=0}^{n}i\cdot a^i$  using summation by parts?  If so, how is it done?  Thanks in advance. What role does "spontaneously symmetry breaking" played in the "Higgs Mechanism"? In talking about Higgs mechanism, the first part is always some introduction to the concept of spontaneously  symmetry breaking (SSB), some people saying that Higgs mechanism is the results of SSB of local gauge symmetry, some people says that we can formulate Higgs mechanism in a gauge invariant way, some people also says that we need only a non-zero vaccum expectation value... I am confused about this different or maybe same point of views.  In this post:  , the most highly voted answer, I still can't feel how SSB worked in Higgs mechanism. It seem that the validity of last part, the appearance of a mass term for $A$, is guaranteed if we have a non-zero equilibrium value $\phi_0$ to expand around. I do not see that the requirement that the phase of the field $\phi$ need to be fixed at some particular value to generate mass term. Thus it seems to me it is not true that SSB is really indispensable for Higgs mechanism.  To put it simply:  The spontanously breaking of what is attributed to Higgs mechanism?   local gauge symmetry global symmetry, since breaking of a "gauge symmetry" should not have any effect on physics. In higgs mechanism, the really broken symmetry is a global one. Mathematically, it is similar in looking as fixing a gauge, but one should not think it as a spontanously breakdown of local gauge symmetry. other.   Is SSB really indispensable for Higgs mechanism?    yes, Higgs mechanism is relied on the SSB of some symmetry (above question), the other approches of description eventually has spontanously broke some symmetry. No, the SSB is just one way to describe Higgs mechanism (or even not a complete way), what is really need is the non-zero vaccum expectation value, for example in the linked post the requirement for the mass term to occur is to have some non-zero expectation value of $\phi$ to expand around, we do not need the phase of the field to be fixed, thus the symmetry is not broken. Other.   some reference materials:    States that SSB of local gauge symmetry is impossible.  in the abstract states that:       gauge symmetries merely reflect a redundancy in the state description   and therefore the spontaneous breaking can not be an essential   ingredient. Indeed, as already shown by Higgs and Kibble, the   mechanism can be explained in terms of gauge invariant variables,   without invoking spontaneous symmetry breaking     In the introduction it says:      In particular, we emphasize that global U(1) phase rotation symmetry,   and not gauge symmetry, is spontaneously violated, and show that the   BCS wave function is, contrary to claims in the literature, fully   gauge invariant Enabling GPU rendering for Cycles? I want to enable GPU rendering, but there is no option in User Preferences > System:    Why is this? How can I get cycles to render using my GPU? CW complex such that action induces action of group ring on cellular chain complex. Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ on the space $\overline{X}$ given by the isomorphism of $\pi$ with $\text{Aut}(\overline{X})$. Let $A$ be an abelian group and let $\mathbb{Z}[\pi]$ act trivially on $A$, $a \cdot \sigma = a$ for $\sigma \in \pi$ and $a \in A$.  Now, assume that $X$ is a CW complex. How do I see that $\overline{X}$ is a CW complex such that the action of $\pi$ on $\overline{X}$ induces an action of the group ring $\mathbb{Z}[\pi]$ on the cellular chain complex $C_*(\overline{X})$ such that each $C_q(\overline{X})$ is a free $\mathbb{Z}[\pi]$-module and$$C_*(X; A) \cong A \otimes_{\mathbb{Z}[\pi]} C_*(\overline{X})?$$ My problem is that, I can not give name of my folder or file with special characters like \ / : * ? " &lt; > | in windows based O.S. This is very important for me, that giving name of file/folder which contain this type of characters.  I already search and refer so many site,for this problems but all are hopeless. I also read Microsoft support page, they suggest use other characters. This is its link,    I already try Alt button + ASCII code number of this type of special characters, from numpad key, just example rename file/folder and press Alt+60 (for &lt; characters) Still it giving message that, A filename cannot contain any of the following characters: \ / : * ? " &lt; > |   Suggest me suitable solution so that I can give name of my file/folder with this type of special characters. Limit values of struct member Python and regular expression with Unicode I need to delete some Unicode symbols from the string 'بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ'  I know they exist here for sure. I tried:  re.sub('([\u064B-\u0652\u06D4\u0670\u0674\u06D5-\u06ED]+)', '', 'بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ')   but it doesn't work. String stays the same. What am I doing wrong? Prove that if every node in a simple graph $G$ has degree $3$ or higher, then $G$ contains a cycle with a chord. Wifi losing connection, weak signal, Intel 7260 adapter I yesterday upgraded from Windows to Ubuntu 15.04 but I have problem with my wireless adapter Intel Corporation Wireless 7260 (rev 73). It randomly loses internet connection. Restarting adapter always helps. (same it did with Windows, except that sometimes restarting adapter didn't help).   Additionally, I have weaker connection then I had with Windows.  Thank you! Let $f(z)$ be entire function. Show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function using Maximum Modulus theorem Let $f(z)$ be entire function. Consider the functions $e^{if(z)}$ and $e^{−if(z)}$ and applying the Maximum Modulus Theorem, show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function.  (We take $f(z)=u(z)+iv(z)$)  I am confused as so far I have $|g(z)|=|e^{if(z)}|=|e^{-v(z)}|$ and then since $f(z)$ is real, $f(z)=u(z)$ and $v(z)=0$ so I assumed it would follow that $|g(z)|=|e^{v(z)}|=1$.  Similarly, $|g(z)|=|e^{-if(z)}|=|e^{v(z)}|=1$.  Using Liouville I assumed one could say that both $g(z)$ and $h(z)$ are bounded entire functions, they are constant and so it follows that $v(z)$ is constant, meaning that both its partial derivatives are equal to 0 and, due to Cauchy Riemann, both of the partial derivatives of $u(z)$ are equal to zero.  It would then follow that $f(z)$ is constant.  I don't know how to go about the question using the Maximum Modulus Theorem, also I feel I am overlooking the importance of $|z|=1$ perhaps?  Any help would be much appreciated!! I am reading separable extension. I don't quite understand why $\text{char} 0$ fields must be perfect. If $F$ is a $\text{char} 0$ field, $f(x)$ is an irreducible polynomial in $F[x]$. Then why $f$ must be separable? Assume $f$ has multiple roots, say $r$ is a multiple root of $f(x)$. Then $r$ is also a root of $f'(x)$. $r$ is in the splitting field of $f(x)$ over $F$, denote it as $E$. Then why $f$ must divide $f'$? I know $f$ is irreducible but I don't know how the argument goes on. Thanks for any help! I know that the rotation period of the moon equals its revolution period. It's just so astonishing that these 2 values have such a small difference. I mean, what is the probability of these 2 values to be practically the same? I don't believe this to be a coincidence. It's just too much for a coincidence. What could have caused this? Before I start, YES, I have done my own research - I know how annoying a repeat question is. Yet each answer is so specific to others...  When I tried the beta 2.8 64 bit, it says "Blender requires a driver with at least OpenGL 3.3 support. The program will now close."    The question: I know that it is true that 2.8 requires OpenGL 3.3. But does my PC not have it, doesn't support it, or do I have it and blender won't detect it? I'm not sure what info to include or what to look for, so here's my PC's specs:  Windows 10 (64 bit) Core -        Intel Core i5-2400 CPU @ 3.10GHz RAM -         8GB (7.89GB usable) System type - 64-bit Operating System, x64-based processor  Thanks!  EDIT: Graphics properties: