I'd like to solve this equation:  $$ u_t  = u_{xx} + u_x  $$ for $t&gt;0$ with initial condition: $$ u(0,x) = \cos (2 \pi x) $$ And $1$-periodic, i.e we have that $u(t,x) = u(t,x+1).$ I was going to solve it using separation of variables, but I was thinking another way, namely solving it using a fourier series. I.e, first I assume that $u$ has fourier series expansion: $$ u(t,x) = \sum_{k \in \mathbb{Z}} a_k(t)e^{2\pi ikx} $$ Now, I would like to differentiate the coefficients in the series and set them equal to each other. Something akin to: $$ a_k'(t) = 2\pi ike^{2\pi i k x} ak(t) + (2 \pi ik )^2 ak(t) e^{2 \pi i k x} $$ And then solve for $ak(t)$. Two questions: 1) is this method valid? I.e is it true that the series are equal if and only if the coefficients are? 2) If I can use this method, how do I solve the ODE for $a_k(t)$? Apologies if this is particularly obvious, I have never taken an ODEs class
For $u:\mathbb{R}\times [0,1]$ with boundary conditions $u(0,x)=\cos (2\pi x)$ and $u(t,0)=u(t,1)$.   Solve $u_t=u_{xx}+u_x$.  I had this on an exam and tried to write $u$ as a product of two single variate functions and convert to an ode using the usual methods but things got messy and I couldn't finish. I also tried to write $u$ as a Fourier series in $x$ with coefficients $a_k(t)$ depending on $t$, but this also didn't seem to yield anything.  Is there a more clever way to approach this, possibly with Fourier series? I heard from a friend there may be a Fourier transform method since the equation was "homogeneous in momentum space."   Thanks!
Don't include SE site in SE search filter? I couldn't find this question here, but I would like to be able to search all sites in the SE network except some sites. In my case, I want to remove the thousands of results from SO on a popular project for devs so I can see the handful of results on other sites from admins and users.  The current form only allows all sites or listed sites.  The answer linked does not tell me how to use the information in the answer. It is a poor answer for an average user.  As @Laurel describes in the comment below, the answer at  no longer works. I'm making a mini game (bomb squad) and the armor needs to be enchanted. I know that armor can be enchanted through command blocks and that is what I intend to do. I am wondering if it is possible to enchant armor twice in one command block or should I need multiple command blocks as I have done in the past? How can I best describe "making a new discovery"? Is $\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$? Getting Started to Xetex for Devangari? The series had three books.  A universe of Greek gods/goddesses, set in modern time.    The gods steal Titans and force them to assume the form of infants/children, and raise them as mundanes in a boarding school type situation.  The protagonist is a young woman who discovers she can travel through the fourth and fifth dimensions as her power.  Her companions: a fey, can travel by unseen ways to various locations; a boy-shaman raised under the traditions of science, can cause anything he wants to happen simply by wishing it and tries to recreate the Michelson–Morley experiment. She ultimately discovers the true nature of the universe she is in and that the deities all follow some sort of rock-papers-scissors system where each deity has a corresponding type which he is weak to, and strong against.  What is the name of this series? Does anyone know any example that invalidates the following affirmation:     If a morphism $f:A\to A$ induces the identity $\hat f:\operatorname{Spec} \left( A \right) \to \operatorname{Spec} \left( A \right)$ then $f = \operatorname{id} _A $. If I select a feature and run self.iface.actionIdentify().trigger() it will open the identify window but won't open the selected feature's data, how can I make it open the data of the selected feature? I am generating a fairly large index of .txt, .tex, and .pdf files. This process has worked for a long time, and the size of the index is not the issue. I have added logic to insert additional links in the index and now am running into:  TeX capacity exceeded, sorry [save size=80000]  Questions:  What are the typical causes of this kind of problem, and any suggestions on how to determine the cause of this. Can the save size be increased?  Of course any solution that just allows me to increase the save size would be preferable if it works, the first question is only important if the size can't be increased or I run into this problem again. Notes:  Generating smaller indices seems to work just fine, only when I attempt to generate the entire index do I run into problems, so coming up with a MWE will be difficult. regex to match word boundary beginning with special characters "Real" cardinality, say, $\aleph_\pi$? Is there any meaningful definition to afford for $\aleph_r$ (as in cardinality) where $r\in\mathbb{R}^+$? $r\in\mathbb{C}$? What about $\aleph_{\aleph_0}$? Can we iterate this? $\aleph_{\aleph_{\aleph_{\cdots}}}$  I may be throwing in bunch of rather naive/basic questions, for I haven't learnt much about infinite cardinalities. If I am referring to bunch of stuff abundantly dealt in established areas, please kindly point out. Shell script zips a folder, but exclude parameters not working IF its run from cron.daily I am solving a clustering problem on a trivial dataset with the k-means algorithm. I am running a couple of tests and, from time to time, one centroid doesn't attract any points, i.e. I've got an empty cluster (see the purple "x" in the picture).  What should I do? Shall I delete it or just stop updating its value? Why?    I am aware that built-in functions (e.g., kmeans() in R) have automatic ways of dealing with this situation, but I am trying to write the standard algorithm from scratch. As soon as I fix it I'll be able to compare my results to built-in functions. At this moment I'm looking for some theoretical reasons why I should prefer one solution or another.  In the picture each colour represents a cluster according to the current iteration and each "X" is its centroid (old ones have been kept and marked with the number of the iteration in which they were computed).      [Some related links about empty cluster case, added later by @ttnphns: ; ; ; ; ; ; ; ; ] biblatex: Cite commands with the same author prints authors name twice I'm using biblatex 1.4b, and Biber 0.9. However, I think the problem is with my settings, not with the versions of those programs.  I'm trying to cite a single author, and produce their name, followed by the dates of two papers, using the authoryear-ibid style, and natbib compatability. Here's my biblatex preamble:  \usepackage[style=authoryear-ibid,natbib=true,backend=biber]{biblatex}   What I want is for the following command: The works by \citet{Blau81, Blau88} ... to produce: The works by Blau (1981; 1988) ... however, what I'm getting is: The works by Blau (1981); Blau (1988)...  I know that biblatex knows they're the same author, since in the bibliography they fall under the same name (i.e. after the first instance of the name, there's just a dash instead of the name)  What am I doing wrong here? I've also tried it with the commands \cite and \cites, and neither produces what I want.   (crossposted with comp.text.tex) I cant change to faces or edges view! using $$ in a post causes strange number series to appear I need a step-by-step, simple and easy way to configure swappiness. Why would I want to install a snap if I can install via apt instead? How do you do load testing and capacity planning for web sites? This is a  about capacity planning for web sites. Related:      What are some recommended tools and methods of capacity planning for web sites and web-applications? Please feel free to describe different tools and techniques for different web-servers, frameworks, etc., as well as best-practices that apply to web servers in general. I'm using an old iBook G4 (PowerPC) running Tiger to serve iTunes to the house. For the past week or so, the mac boots fine, and then sometime during the day starts generating   An unexpected error occurred (error code -50).   errors when iTunes attempts to save its library file.  Checking reveals that the boot drive (main drive, in the laptop) behaves as if it is write-protected - I get the same error code when I attempt to create a folder on the desktop, for instance. If I attempt to create files via the terminal, e.g. via     ls > foo.txt I get an "invalid argument" error.   I've set the iBook to run in Target Disk Mode and run Disk Utility to repair the disk (from a Macbook running Snow Leopard) and get "The volume appears to be ok".   I don't have the option in Disk Utility to Repair Permissions in Target Disk Mode. When I ran Repair Permissions on the iBook itself, it came back with no errors, no permissions corrected, but did not in fact fix the problem.  Rebooting always solve the problem - for the next 5-10 hours or so.   Would Disk Warrior help, or some other software fix, or is it time to buy a new drive? (And, in any case, what's going wrong?)