This question must be weird, but:     Is $f(x) = 1$ equal to $f(x) = \frac{x}{x}$ ?   The reason I am asking is:  Since $\frac{x}{x} = 1$, it would be reasonable to conclude $f(x) = 1$ is indeed equal to $f(x) = \frac{x}{x}$. However, it we look at the graphs of these two functions, we can see that $f(x) = 1$ is continuous everywhere, yet $f(x) = \frac{x}{x}$ is undefined at zero.
I understand that division by zero isn't allowed, but we merely just multiplied $f(x) = 1$ by $\frac{x-1}{x-1}$ to get $f(x) = \frac{x-1}{x-1}$ and $a\cdot 1 = 1\cdot a = a$ so they're the same function but with different domain how is this possible?  Or in other words why don't we simplify $f(x) = \frac{x-1}{x-1}$ to $f(x) = 1$ before plotting the points. Is it just defined this way or is there a particular reason ?  Note: my book says the domain of $f(x) = 1$ is $\mathbb{R}$ and the domain of  $f(x) = \frac{x-1}{x-1}$ is $\mathbb{R}$ except $1$.
Confirmation that someone is listening to another person's speech When someone is telling you a very long and detailed story he usually wants to hear some "confirmations" (or response) that you are listening to his story. In Russian we often use something like "tak" (which has a meaning of "ok" and "well, proceed further"), "a-ha" or "uh-huh".  What word serves the same purpose in English and American English? Get number of frames in scene from the command line? I'd like to know if there is a command-line option (or combination, or a way!) to get the number of frames that are included in a specified blender scene; for example I'd like to use is like this:  $ blender &lt;option&gt; file.blend $ Number of frames in blender scene: xxxx What would be the easiest way to view classes, methods, properties, etc. inside a jar file? I'm looking for something equivalent to the very useful  Lutz Roeder .NET Reflector  - for Java Find weak form of linear transport equation 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. Suppose $R\neq 0$ is a commutative ring with $1$. Let $M$ be a free $R$-module. I would like to prove that $M$ is a flat $R$-module. Everywhere I have looked (mostly online) this is proved by first proving that every free module is projective, and then proving that every projective module is flat. Unfortunately, Atiyah &amp; Macdonald's "Introduction to Commutative Algebra" (Chapter 2) does not discuss projective modules. But the result that every free module is flat comes very handy in the exercises.   So my question is,     Is it possible to prove that every free module is flat just by   definitions and without appealing to projective modules?   Thanks! Why doesn't terminal show color schema once logged as root? I'm currently using Debian 7 Wheezy, and I've noticed that the colors palette on terminal (which might be useful to identify among a large set of files and folders) are not enabled once I log as root.  Same, I've noticed the tab completion doesn't work the same in this situation.  How can I configure these issues? What's the main difference between Intel processor generations? What are the main differences between each processor generation of Intel processors/chipsets? Is there a performance difference?  I've already seen processors from first to sixth generation, however I don't know/understand the exact differences. Repeat the same reference in footnote on different pages What exactly is the cause of RAM incompatibility? I'm looking at rather short QVL lists from manufacturers, and I'm wondering what is the underlying cause of RAM incompatibility aside from the obvious: Obvious Reasons for Incompatibility  RAM Generation (DDR, DDR2, DDR3, DDR4) Form Factor / pin count RAM Type (ECC vs Non-ECC, Registered vs not, Buffered vs not) Minimum speeds on a board that cannot adjust the memory frequency Non-standard voltage range on a board that cannot adjust the memory voltage Channel count (single vs dual vs quad) Bad or inoperable module Mismatched set  Not Obvious Reasons  Brand SKU Max Capacity Speed too high  As in, why can't motherboard manufacturers just list the specifications of the RAM they will accommodate, and support every memory chip of that category instead of listing specific SKUs from specific brands? What happens electrically/physically that causes boards to fail to POST / operate with RAM that is otherwise 'compatible' with all of the specifications of RAM that is on the QVL? How to install software or upgrade from an old unsupported release? Recently I have installed an older version of Ubuntu on my old machine. Whenever I try to install any software, I get an error saying it couldn't be found:  $ sudo apt-get install vlc Reading package lists... Done                Building dependency tree        Reading state information... Done     E: Couldn't find package vlc Boundary conditions for calculus of variations in phase space and under canonical transformations This might be a stupid question, but I just don't get it. In Hamiltonian mechanics when examining conditions for a $(\boldsymbol{q},\boldsymbol{p})\rightarrow(\boldsymbol{Q},\boldsymbol{P})$ transformation to be canonical one starts with $$ \dot{q}_ip^i-H(\boldsymbol{q},\boldsymbol{p},t)= \dot{Q}_iP^i-\bar{H}(\boldsymbol{Q},\boldsymbol{P},t)+\frac{d}{dt}W(\boldsymbol{q},\boldsymbol{Q},t)$$ where $\bar{H}$ is the transformed Hamiltonian, and $W$ is the generating function (now a function of $\boldsymbol{q}$ and $\boldsymbol{Q}$). This term shouldn't break Hamilton's principle, since $$ \delta\int_{t_1}^{t_2} dt\frac{d}{dt}W(\boldsymbol{q},\boldsymbol{Q},t)=\delta W(\boldsymbol{q},\boldsymbol{Q},t)|_{t_2}-\delta W(\boldsymbol{q},\boldsymbol{Q},t)|_{t_1}=0-0=0 .$$ But I don't see why the variation of $W$ should disappear at the endpoints (say at $t_1$). Expanding leads to: $$ \delta W(\boldsymbol{q},\boldsymbol{Q},t)|_{t_1}=\left(\frac{\partial W}{\partial q_i}\right)_{t_1}\underbrace{\delta q_i(t_1)}_{=0}+ \left(\frac{\partial W}{\partial Q_i}\right)_{t_1}\delta Q_i(t_1)=\left(\frac{\partial W}{\partial Q_i}\right)_{t_1}\delta Q_i(t_1).$$ $\boldsymbol{Q}$ is itself a function of $\boldsymbol{q}$ and $\boldsymbol{p}$, so $$ \delta Q_i(t_1)=\left(\frac{\partial Q_i}{\partial q_k}\right)_{t_1}\underbrace{\delta q_k(t_1)}_{=0}+\left(\frac{\partial Q_i}{\partial p_k}\right)_{t_1}\delta p_k(t_1)=\left(\frac{\partial Q_i}{\partial p_k}\right)_{t_1}\delta p_k(t_1). $$ It seems as if we also needed the variation of $\boldsymbol{p}$ to vanish at the endpoints, and I don't get this because (at least in cartesian coordinates) $\boldsymbol{p}=m\dot{\boldsymbol{q}}$ and the velocity can be different along the original and the varied orbitals even at the endpoints (they can point in totally different directions), so in general $\delta \dot{\boldsymbol{q}}(t_1)\neq 0$. What am I doing wrong? Can someone help me with this, please? all the LaTeX-tutorials tell me how LaTeX is an extention simplifying the building of TeX documents. Understandably, no one uses plain TeX anymore, right?  But I seem to learn better when I grasp the link between TeX and LaTeX. Does someone have a link to example files in plain TeX? I'd like to know which commands, macros etc. are LaTeX-specific. Why do some questions not start with an auxiliary verb? When I learned English, my teachers told me that all questions must have an auxiliary verb at the beginning, just like Are you mad? or Is she playing? do.  But when watching some movies or talking with people who speak English, they just ask using things like You mad? and She’s playing?   Of course, the kind of intonation let me know they are asking a question, but why this happen? Creating a default preamble Since I am using (almost) the same set up for my tex documents, I wonder if there is a way to create a preamble with my default packages and to recall it anytime I want to create a tex document.  If yes, how I can call this "default" preamble?  Thanks in advance What is some good free software for doing physics calculations?  I'm mainly interested in symbolic computation (something like Mathematica, but free). How do I find peaks in a dataset? If I have a data set that produces a graph such as the following, how would I algorithmically determine the x-values of the peaks shown (in this case three of them): When I try to remove Java with sudo apt-get remove jdk1.8.0-40 I get this:    Reading package lists... Done Building dependency tree        Reading state information... Done The following packages will be REMOVED:   jdk1.8.0-40 0 upgraded, 0 newly installed, 1 to remove and 0 not upgraded. 1 not fully installed or removed. After this operation, 245 MB disk space will be freed. Do you want to continue? [Y/n] y (Reading database ... 231488 files and directories currently installed.) Removing jdk1.8.0-40 (1.8.040-1) ... find: `/usr/java/*': No such file or directory /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 586: /usr/sbin/alternatives: No such file or directory /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 598: /usr/sbin/alternatives: No such file or directory dpkg: error processing package jdk1.8.0-40 (--remove):  subprocess installed post-removal script returned error exit status 127 Errors were encountered while processing:  jdk1.8.0-40 E: Sub-process /usr/bin/dpkg returned an error code (1)   Output of  ls -1 /var/cache/apt/archives/*jdk*:  openjdk-7-jre_7u75-2.5.4-1~trusty1_amd64.deb openjdk-7-jre-headless_7u75-2.5.4-1~trusty1_amd64.deb   Output of sudo dpkg --purge --force-all jdk1.8.0-40:  (Reading database ... 231488 files and directories currently installed.) removing jdk1.8.0-40 (1.8.040-1) ... /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 586: /usr/sbin/alternatives: No such file or directory /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 598: /usr/sbin/alternatives: No such file or directory dpkg: error processing package jdk1.8.0-40 (--purge): subprocess installed post-removal script returned error exit status 127 Errors were encountered while processing:  jdk1.8.0-40   When trying the edit I get this:  (Reading database ... 231488 files and directories currently installed.) Removing jdk1.8.0-40 (1.8.040-1) ... /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 586: /usr/sbin/alternatives: Permission denied /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 598: /usr/sbin/alternatives: Permission denied dpkg: error processing package jdk1.8.0-40 (--remove):  subprocess installed post-removal script returned error exit status 126 Errors were encountered while processing:  jdk1.8.0-40 E: Sub-process /usr/bin/dpkg returned an error code (1) 'Which', 'whose' or something else? This question was phrased as a bug report for LaTeX and I thought it would be helpful to be presented here (including an answer).  The person had a lot of "common words" in typewriter font sprinkled around his text and he was surprised that this text would not automatically hyphenate and he had to manually insert \- to mark hypenation points. How to install Epson Perfection V600 Photo Scanner drivers