Let $T:C([0,1])\rightarrow C([0,1])$ defined by  $Tf (t) = f(t) + \int_{0}^{t} f(s) ds$,  It is easy to show that $T$ is a linear and continous operator, but I'm stuck about showing injective,surjective or exhibiting its inverse. How can i compute those things?
Define a linear operator $T:C[0,1] \to C[0,1]$ as follows: $$Tf(x) = f(x) + \int_0^x f(u)du$$ It is easy to show that $T$ is a bounded linear operator. The statement also (1) claims that $T$ is one-to-one and onto, and (2) asks to compute $T^{-1}$. (Hence $T$ is an isomorphism due to a corollary of Open Mapping Theorem)  I need help to prove $T$ is onto: given any $g \in C[0,1]$, how to decompose it into a sum between a continuous $f$ and its antiderivative. I have trouble decomposing $g(x)=|x|$, not to mention bizarre functions like Weierstrass function. It seems impossible.  Thanks in advance for help!
How does a fork bomb work? WARNING DO NOT ATTEMPT TO RUN THIS ON A PRODUCTION MACHINE   In reading the  I generally follow what's going on with the following code:  :(){ :|:&amp; };:   excerpt of description     The following fork bomb was presented as art in 2002;   its exact origin is unknown, but it existed on Usenet prior to 2002.   The bomb is executed by pasting the following 13 characters into a    shell such as  or . It operates by defining   a function called ':', which calls itself twice, once in the   foreground and once in the background.   However the last bit isn't entirely clear to me. I see the function definition:  :(){ ... }   But what else is going on?  Also do other shells such as ksh, csh, and tcsh also suffer the same fate of being able to construct something similar? Example:  A tall(,) green pole.   says that cumulative adjectives don't need comma. In my example, &quot;tall&quot; describes shape, and &quot;green&quot; describes color. So they are cumulative adjectives. But  says that &quot;tall&quot; and &quot;green&quot; are equivalent and coordinate adjectives ... I'm confused. Should there be a comma or not? What are the ways to position things absolutely on the page? I know that there is tikz, but also eso-pic and textpos.  I'd like to see a list of possible ways (not necessarily packages) for absolute positioning of things (which I believe amounts to "boxes") on the page, together with their advantages and disadvantages (for instance: tikz's current page node is very easy to use, but usually requires two LaTeX runs, and overhead from loading tikz if you don't need it for any other things is considerable).  Both LaTeX and ConTeXt solutions are welcome I'm confused. Running Fedora Linux, lscpu yields:  Architecture:            i686 CPU op-mode(s):          32-bit, 64-bit ...   But when I try to install a 64-bit program (Chrome) I get error like:     Package /....x86_64.rpm has incompatible architecture x86_64. Valid architectures are ['i686', 'i586', 'i486', i386']   I'm less interested in being able to install Chrome and more interested in why lscpu says that my CPU can run in 64-bit mode; clearly this can't mean I can run 64-bit programs. Can anyone clarify? How can we recover the Newtonian gravitational potential from the metric of general relativity? The Newtonian description of gravity can be formulated in terms of a potential function $\phi$ whose partial derivatives give the acceleration:  $$\frac{d^2\vec{x}}{dt^2}=\vec{g}=-\vec{\nabla}\phi(x)=\left(\frac{\partial\phi}{\partial x}\hat{x}+\frac{\partial\phi}{\partial y}\hat{y}+\frac{\partial\phi}{\partial z}\hat{z}\right)$$  However, in general relativity, we describe gravity by means of the metric. This description is radically different from the Newtonian one, and I don't see how we can recover the latter from the former. Could someone explain how we can obtain the Newtonian potential from general relativity, starting from the metric $g_{\mu\nu}$? utf8 and ISO 8859-1 encoding references trouble I have to use one particular template ( am obliged to use it for my thesis) it uses \bibliographystyle{plain} Ihave both english and frensh text on it (for abstract and résumé) there is no problem for characters like ( é, ç...) but some of references does not appear they are replaced by [?] knowing that foe encoding \RequirePackage[latin1]{inputenc} how to fix this problem please!   \documentclass{article}   \RequirePackage[latin1]{inputenc}   \RequirePackage{fancyhdr}     blabla \cite{nayak2015comprehensive, byun2002applications, shah2004sequential, fletcher2009support, cornuejols2011apprentissage, byun2002applications}   @article{nayak2015comprehensive,    title={A comprehensive survey on support vector machine in data mining     tasks: applications \&amp; challenges},    author={Nayak, Janmenjoy and Naik, Bighnaraj and Behera, HS},    journal={International Journal of Database Theory and Application},    volume={8},    number={1},    pages={169--186},    year={2015}    }      @inproceedings{shah2004sequential,     title={Sequential k-nearest neighbor pattern recognition for usable      speech classification},     author={Shah, Jashmin K and Smolenski, Brett Y and Yantorno, Robert E      and Iyer, Ananth N},     booktitle={Signal Processing Conference, 2004 12th European},     pages={741--744},     year={2004},     organization={IEEE}     }      @misc{fletcher2009support,      title  = {Support vector machines explained},     author = {Fletcher, Tristan},     journal= {University College London, London},     year   = {2009},     howpublished = "\url{http://sutikno.blog.undip.ac.id/files/2011/11/SVM-     Explained.pdf}",     note   = {Accessed: 2017-04-30}     }      @inbook{cornuejols2011apprentissage,      title={Apprentissage artificiel: concepts et algorithmes},      author={Cornu{\'e}jols, Antoine and Miclet, Laurent},      year={2011},      pages={279--309},      publisher={Editions Eyrolles}       }      @incollection{byun2002applications,      title={Applications of support vector machines for pattern recognition:       A survey},      author={Byun, Hyeran and Lee, Seong-Whan},      booktitle={Pattern recognition with support vector machines},       pages={213--236},      year={2002},      publisher={Springer}      }    \end{document}  knowing that my references work well on other template but this one no how to fix this problm please! here is the error I got   Edited here is the link to the template  in fact I have created a littele example ( since I could not publish my thesis ...) the error has disappeared but my references still [?] Do I have to create vector files at the correct print size? Migrated question's answers can be upvoted twice by the same user Why $¬¬\bot \not\in PROP$? I'm reading Dalen's Logic and Structure.  He gives the definition of the set $PROP$:     Definition $\bf2.1.2$ The set $PROP$ of propositions is the smallest set $X$ with the properties       $\begin{array}{rl} \rm(i)&amp;p_i\in X(i\in N),\bot\in X,\\ \rm(ii)&amp;\varphi,\psi\in X\Rightarrow(\varphi\wedge\psi),(\varphi\vee\psi),(\varphi\to\psi),(\varphi\leftrightarrow\psi)\in X,\\ \rm(iii)&amp;\varphi\in X\Rightarrow(\neg\varphi)\in X.\\ \end{array}$   Then there are some examples:        And he says that $¬¬\bot \not\in PROP$, but I don't understand it. Isn't $¬¬  \bot=\bot$ due to the property of double negation? Considering this is true, $\bot\in PROP$. Of course, in this question, I'm pressuposing some kind of evaluation, that $¬¬ \bot$ becomes $\bot$ due to the law of the double negation, but perhaps it should be hold in an unevaluated form.  Ex:. $1+1=2$ because we presupose the $1$ merges with $1$ and then becomes two. If the merge (evaluation) does not happen, then I guess:  $$1+1=1+1$$ $$2=2$$ $$2\neq 1+1$$ $$1+1\neq 2$$  Perhaps it's something like that, I took the idea from Mathematica. Sometimes it does treats things as different in the absence of full evaluation. English questions and negation with *do* in syntax Prove by induction that... $1+3+5+7+...+(2n+1)=(n+1)^2$ for every $n \in \mathbb N$ What does the size of the standard deviation mean? According to  (my emphasis),     In statistics, the standard deviation ... is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A standard deviation close to 0 indicates that the data points tend to be very close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.   The purpose of the standard deviation (SD), then, is to tell us how varied or uniform (SD → 0) the data is.  So, given a certain SD, how varied is the data? What does the size of the standard deviation mean?  Please explain the meaning of the SD by interpreting an SD = 1 (M = 0).  If you cannot interpret the size (quantity) of this SD, what other information would you need to be able to interpret it, and how would you interpret it, given that information? Please provide an example.  If, on the other hand, the quantity of the SD cannot be qualified in this manner, my argument is that it is essentially meaningless. If you disagree, please explain the meaning of the SD.  &nbsp;    The following are earlier versions to give context to the answers. Unfortunately these didn't really convey what I wanted, and my attempt to ask it elsewhere was closed. (I don't need these versions answered now):  First revision:  What does the size of the standard deviation mean?  For example, if I want to study human body size and I find that adult human body size has a standard deviation of 2 cm, I would probably infer that adult human body size is very uniform, while a 2 cm standard deviation in the size of mice would mean that mice differ surprisingly much in body size.  Obviously the meaning of the standard deviation is its relation to the mean, and a standard deviation around a tenth of the mean is unremarkable (e.g. for IQ: SD = 0.15 * M).  But what is considered "small" and what is "large", when it comes to the relation between standard deviation and mean? Are there guidelines similar to the ones that Cohen gives for correlations (a correlation of 0.5 is large, 0.3 is moderate, and 0.1 is small)?    Original question:  We always calculate and report means and standard deviations. But what does the size of the variance actually mean?  For example, assume we are observing which seat people take in an empty room. If we observe that the majority of people sit close to the window with little variance, we can assume this to mean that people generally prefer siting near the window and getting a view or enough light is the main motivating factor in choosing a seat. If on the other hand we observe that while the largest proportion sit close to the window there is a large variance with other seats taken often also (e.g. many sit close to the door, others sit close to the water dispenser or the newspapers), we might assume that while many people prefer to sit close to the window, there seem to be more factors than light or view that influence choice of seating and differing preferences in different people.  At what values can we say that the behavior we have observed is very varied (different people like to sit in different places)? And when can we infer that behavior is mostly uniform (everyone likes to sit at the window) and the little variation our data shows is mostly a result of random effects or confounding variables (dirt on one chair, the sun having moved and more shade in the back, etc.)?  Are there guidelines for assessing the magnitude of variance in data, similar to Cohen's guidelines for interpreting effect size (a correlation of 0.5 is large, 0.3 is moderate, and 0.1 is small)?  For example, if 90% (or only 30%) of observations fall within one standard deviation from the mean, is that uncommon or completely unremarkable? How to prove that $p-1$ is squarefree infinitely often? How to prove that $p-1$ is squarefree infinitely often, where $p$ is prime?  I had thought of using Dirichlet's theorem on arithmetic progressions.  The number of squareful values of $p-1$ less than or equal to $n$ is bounded by $\sum_{q}{\pi_{q^2,1}(n)}$, each term $\pi_{q^2,1}(n) \approx \frac{\pi(n)}{q \cdot (q-1)}$, and $\sum_{q}{\frac{1}{q \cdot (q-1)}} \lt 1$.  But using for example , the possibility is open that the error in each approximation for $\pi_{q^2,1}(n)$ may continue to increase with $q$, and since the sum runs up to $\sqrt{n}$, this argument is invalid.  Is there a more refined version of Dirichlet's theorem that can be used to circumvent this issue?  Or an entirely different way to prove this statement? Did Snape ever commit murder? It's well known that Severus Snape was, in one way or another, a Death Eater since he was at Hogwarts. He hung out with the likes of Avery and Mulciber who became well known Death Eaters, he joined Voldemort right after he left Hogwarts, and he basically became Voldemort's right hand man until the very end.   However, in his time as a Death Eater did he ever kill anyone? We know that he was a Death Eater for around 3 years before Voldemort fell; Lily and James died at 21 and they were the same age as Snape, so it would have been 3 years since they left Hogwarts. Then after Voldemort came back in book 4, Snape would have been a Death Eater again for around 3 years. That's a total of around 6 years of Death Eaterhood with Voldemort.  So, in either of these periods, did Snape commit murder as part of his job as Death Eater? I'm not counting a certain death at the end of book 6 for reasons which are obvious if you've read book 7.   Clearly it's much more likely he killed before Voldemort fell since that roughly correlated with the period when he pledged allegiance only to Voldemort, before he became a teacher at Hogwarts. But did he ever do it? MySQL single table static and dynamic pivot I have a table that looks like this:  +----------------------------------------+ |Name                | kode      | jum   | +----------------------------------------+ | aman               |kode1      | 2     | | aman               |kode2      | 1     | | jhon               |kode1      | 4     | | amir               |kode2      | 4     | +--------------------+-----------+-------+   How can I make a view like this with MySQL?                      kode1    kode2     count aman                  2         1        3   jhon                  0         4        4 amir                  0         4        4 Eigenvalues of a matrix whose square equals its transpose Let $A$ be a $n\times n$ matrix with $A^2=A^t$. Show that every real eigen value of $A$ is either $ 0$ or $1$ Passing multiple commands to gnome-terminal from a script My bash-script is not run in a terminal, but opens a terminal at runtime to ask for user input (sudo password). I would like to tell the user in this terminal what the script is trying to do. First, here is my script:  #!/bin/bash #This should install: Qt Linguist, lrelease  #for Qt Linguist dpkg-query -l qt4-dev-tools if echo $? == 0; then  check1=1 fi  #for lrelease dpkg-query -l qt4-default if echo $? == 0; then  check2=1 fi  echo $check1 echo $check2  if [ $check1 = 1 ] &amp;&amp; [ $check2 = 1 ];then  gnome-terminal -x sudo apt-get install qt4-dev-tools qt4-default fi $SHELL   This opens a terminal with this:  [sudo] password for "username":    But I would like to have:  qt4-dev-tools and qt4-default are missing. Now trying to install. [sudo] password for "username":    How do I get this when starting the terminal with gnome-terminal -x? From injective map to continuous map So I understand that this has happened in the past: , but all of the &quot;duplicate&quot; posts are marked as  so here is a new bug report: On the , I am listed twice:  While that's fun and all, it isn't exactly right. I've been using SSH Tunnel manager to easily set up and stop port tunneling over SSH to various servers at my company. That saved me the pain of opening up a new terminal window which will hang out there just for the sake of having a tunnel open.  This worked great (well , sort of) but SSH Tunnel Manager is a PowerPC application. With Lion, it is not supported any more.  What is a good replacement?