Let $v \in L^1_\text{loc}(\mathbb{R})$ weakly differentiable, $b\in \mathbb{R} \setminus \{0\} $ constant and $u:\mathbb{R}^2 \to \mathbb{R}$ defined by $u(x,t):= v(x-bt)$.  Show that $u$ has weak derivatives $ \partial_x u$ and $ \partial_t u$ which satisfy the transport equation $\partial_t u + b \partial_x u = 0$ almost everywhere.  I know the definition of the weak derivative, but since I don´t have any explicit function given I don´t know how to test it. I also wonder if there is a chain rule for weak derivatives, since $u$ is a concatenation of $v$ with $x-bt$.   Could anyone help me out with this problem? Thank you very much in advance!
I am stuck on the following problem that says:       a) Find a weak formulation for the partial differential equation $${\partial u\over\partial t\ }+ c{\partial u \over \partial x\ }=0$$    b) Show that $u=f(x-ct)$ is a generalized solution of $${\partial u\over\partial t\ }+ c{\partial u \over \partial x\ }=0$$ for any distribution $f$    My Attempt:   I know that in order to find a weak form of a pde, we need to multiply it by a test function, then integrate it. Also, to find a generalized solution, we need to find a weak solution and just multiply it by the Heaviside function.  Let's take any test function $\phi $, then we have (integrating by parts second part of the integral) $$\int_{\Omega } ( {\partial u\over\partial t\ }+ c{\partial u \over \partial x\ })*\phi(x) dx= \int_{\Omega } {\partial u\over \partial t}\phi(x) dx - c\int_{\Omega } u(x,t)\phi'(x)dx$$ where $\phi $ vanishes at boundaries. So, is it the final form or can we proceed further? And how am I supposed to find a generalized solution?  Can someone help me out? Thanks in advance for your time.
Since the final Black Hole (BH) had 3 solar masses less of mass than the original binary BH, it seems the 2 BHs lost mass, and with it event surface area, entropy, and information. If that came from inside the BHs it seems that it would be a contradiction of what a BH is --'nothing escapes its event horizon'. Some may say that came from the rotational energy of the pair, but the new BH now seems to have less than the two original ones summed. So, did some of that escape the horizons? How do I set up my LaTeX document to enable me to cite from my BibTeX bibliography? This may be an overly simplistic question but please bear with me...  I have a BibTeX bibliography containing all of my references, set up using BibDesk. I am now trying to cite these references in my LaTeX documents. Following the advice on the LaTeX wikibooks page, I have included the following lines into my LaTeX document...  \bibliographystyle{plain} \bibliography{/Users/Graeme/Documents/References/BIBliography.bib}   ... where BIBliography is my .bib file, created in BibDesk.  Unfortunately, when I cite something in the document, using \cite{citekey} and typeset the document, the log tells me that there were undefined references. In the .pdf that is produced, there is a [?] where the references should be.  Am I doing something wrong? Do I need to store my .bib for in a particular location? I know there can be some errors that mean you need to typeset the document several times, but I have tried countless times to no avail.  I would be very grateful if someone could help out. Beamer how to change color of infolines and frame title How could I change the color of color of infolines and frame title in beamer. I am using this now   \documentclass[xcolor=dvipsnames]{beamer} \usetheme{CambridgeUS} \useinnertheme{rectangles} \useoutertheme{infolines} \usecolortheme[named=Brown]{structure}    I need to change the color from red to brown. [named=Brown] doesn't work for \useoutertheme{infolines}. This is my current output. I also need to change the background of frame title (mouse pointer) Unknown Column In Where Clause I have never had this problem before. But after seeing some questions posted on ELU, I started to doubt. I have been through links1, pages of grammar books2, a lot of questions3 posted here and ELU. I now still can't seem to understand how the past perfect is exactly used. The references sometimes even contradict each other.  What I had understood so far was      The past perfect is not necessary if we are not 'going back' to the earlier past, and simply moving from one event to another.   but I then saw a lot of other 'rules' including      If the sequence is clear, the past perfect is not needed.   which is slightly different, less 'narrow' and can be applied in more situations than the first ruleA, and      The past perfect is optional only when talking about an action at a specific time.   This is confusing. Can anyone here clarify things out please?    A For example, for  Doing the survey [completed] -&gt; Showing the result      We showed the result of the survey we did.      If the second rule is applied, the sentence above is fine (the sequence is clear), but not if the first rule since we are not moving from any event to the next and possibly even going back.    1  and   2 Practical English Usage (2005) by Michael Swan, Collins Cobuild Student's Grammar (1991). Upper-Intermediate Cutting Edge.  3 including , , ,  etc. How to select multiple files with keyboard only in windows explorer I found the below in  web-site:     We use much in questions and negative clauses to talk about degrees of something.      We don’t use much in affirmative clauses.   Also, it the same web-site, I found the following:     She runs much faster than he does.    also I noticed that some web-sites use much with affirmative clauses:     May we all be having this much fun and getting this much exercise when we're his age!    Is it Okey to use much in affirmative clauses? If yes, when can we use it? and what is the rules?  Update:  Here is another example on how often the word "much" is used  in informal style:     After much consideration I will NOT be performing at the Inauguration with my improv group Capitol Hill-arious Subgroup of $\mathbb{R}$ either dense or has a least positive element? Why Process Builder doesn't work on Delete Scenario? With spells like Moonbeam or Flaming Sphere, how should we treat the movement of their areas-of-effect when it comes to passing them behind a thick pillar or around a blind corner?  Should we treat any movement of them as if they are being 're-cast' (and so requiring a clear unobstructed view)? Or could a mage presumably will them somewhere out-of-sight? Collection of short stories; in one a spy attempts to escape detection on Mars's moon by moving faster than the spaceship can scan How to show that the cohomology of a Grassmannian has a basis consisting of the equivalent classes represented by Schubert cycles? Let $G(r, n)$ be the Grassmannian of the set of all $r$-planes in a $n$-dim vector space. How to show that the cohomology of a Grassmannian has a basis consisting of the equivalent classes represented by Schubert cycles? I am confused since I don't know how to compute the cohomology of a variety (how to construct the co-chain complex). 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I recently bought one of these ESR meters, a 'MESR-100':       It claims to be able to test capacitor ESR with the caps left in-circuit.  It appears to work well from my tests so far.  However, how much can I trust the in-circuit readings?  Are they going to be less accurate?  Are there certain circuit configurations where it wouldn't be possible to measure ESR in-circuit accurately?  [EDIT: Added the model as requested] : What is the maximum value of $n$ if $n + 5$ divides $n^5 + 5$, $n$ being a natural number? I tried to solve by binomial theorem but it failed. Cannot start XAMPP on Ubuntu 11.10 Can I upload an entire folder using FTP? I need to upload a full folder using FTP.  Is there is any option for transferring a folder and all of its contents at once? IF statement inside a LINQ SELECT