Q. Let $T: C[0,1] \rightarrow C[0,1]$ be a mapping defined by $$ T f(x)=\int_{0}^{x} f(s) d s $$ Show that: $T$ is NOT a contraction but $T^{2}$ is a contraction. It seems to me that $\begin{aligned}\left|\int_{0}^{x}(f(s)-g(s)) d s\right| &amp; \leq \operatorname{sup}_{[0,1]}|f(s)-g(s)| \int_{0}^{x} d s \\ &amp; \leq \sup _{[0,1]} |f(s)-g(s) |\end{aligned}$ What point I have missed here?
Let $\psi: C([0,1]; \mathbb{R}) \mapsto C([0,1]; \mathbb{R})  $, be such that if $x \in [0,1]$ then $$\psi(f)(x) = \int\limits_{0}^{x} f(t) dt $$  I am asked to show that $\psi$ is not a contraction but $\psi^{2}$ is. And then conclude that $\psi$ has a unique fixed point.   I´ve managed to prove a statment that helps me conclude the last part but I´m having trouble proving that the function is not a contraction. This is what I've got.   If $f,g \in  C([0,1]; \mathbb{R})$  then $$ |\psi (f) - \psi(g)| \leq \int\limits_{0}^{x} |f(t)-g(t)| dt \leq \sup\limits_{x\in [0,1]} |f(x)-g(x)| \int\limits_{0}^{x} 1 dt = d(f,g) x  $$ Then $|\psi (f) - \psi(g)| \leq d(f,g) x $,  and taking supremum on the left hand side allows me to conclude that $d(\psi (f),\psi (g)) \leq d(f,g) x$  Is there any problem with this procedure? Does anybody has an insight on how to procede with the $\psi^{2}$ part?   Many Thanks!
Why does every server I try say it can't connect? I just re-purchased Minecraft (because my old profile stopped working so I bought it) after not playing it for a couple of months. I've added a bunch of servers, but almost every one says "can not connect."  I know its not because the server is not 1.7.2, because I checked that they'd update. It will say "server can't connect" after a long time. I don't know if its me or the servers.  My server screen looks like this (notice the one at the bottom – it's still loading!): Prove $2^b-1$ does not divide $2^a + 1$ for $a,b>2$ I'm trying to prove $2^b-1$ does not divide $2^a + 1$ for $a,b&gt;2$. Can someone give a hint in the right direction for this? Duplicating Environments What is the meaning of the term "thread-safe"? Does it mean that two threads can't change the underlying data simultaneously? Or does it mean that the given code segment will run with predictable results when multiple threads are executing that code segment? I posted a comment to a duplicate question on . But I decided to delete it. Afterwards I voted as duplicate and thought a comment would be generated like     possible duplicate of ...   But it didn't. My guess is that the duplicate-comment won't be generated if a comment exists with that link, even if deleted. I have seen people apply a filter to imported data before plotting with pgfplots, see . Is there a way to extend the approach presented there to apply multiple filters? In the MWE below I made the naive modifications to the solution given there, but it seems that only the last filter is applied: the unwanted y=1 data points are plotted too.  \documentclass{article} \usepackage{filecontents} \usepackage{pgfplots}  \begin{filecontents}{foo.txt}   x y z f   0 0 0 0   0 0 1 1   0 1 0 2   0 1 1 3   1 0 0 4   1 0 1 5   1 1 0 6   1 1 1 7   \end{filecontents}  \pgfplotsset{   discard if not/.style 2 args={   x filter/.code={   \edef\tempa{\thisrow{#1}}   \edef\tempb{#2}   \ifx\tempa\tempb   \else     \def\pgfmathresult{nan}   \fi   }}   }  \begin{document} \begin{figure}   \centering   \begin{tikzpicture}   \begin{axis}   \addplot table[x=x, y=f,                  discard if not={y}{0},                  discard if not={z}{0},                 ] {foo.txt};   \end{axis}   \end{tikzpicture}   \caption{$f(x)$ given $y=0$ and $z=0$.}   \end{figure} \end{document} What are integers $x$ and $n$ such as $ x^4+x^3+x^2+x+1 = n^2$? What are the integer numbers $x$ such as $x^4+x^3+x^2+x+1$ is the square of an integer?  $0$ and $-1$ are obvious answers, but how many are there? "He needn't say" / "He doesn't need to say" When I studied English at school (in Italy, more than 35 years ago), we were taught a rule according to which one should say, e. g., "He needn't (need not) say", but NOT "He doesn't need to say".  If I'm not mistaken, it was explained to us that the rule holds good with the following limitations:  1) only in the negative form: thus, "He needn't say", but "He needs to say", and "Does he need to say?";  2) only in the present tense: thus, "He needn't say", but "He didn't (won't, wouldn't, etc.) need to say";  3) only when an infinitive follows: thus, "He needn't see a doctor", but "He doesn't need a doctor".  But, if I google "doesn't need to", I find 125,000,000 occurrences.  I was wondering: does the rule hold good more for British than for American English?  Also, perhaps it isn't an absolute rule, and both constructions ("He needn't say" and "He doesn't need to say") are possible.  Is the rule at least a good description of the circumstances when we CAN (but needn't necessarily) use this construction?  Thank you very much, for this and the other answers. Writing string at the same position using Console.Write in C# 2.0 I have a console application project in C# 2.0 that needs to write something to the screen in a while loop. I do not want the screen to scroll because using Console.Write or Console.Writeline method will keep displaying text on console screen incremently and thus it starts scrolling.  I want to have the string written at the same position. How can i do this?  Thanks Wrap text around wide figure in two column I need to insert a figure into a two column document such that the figure partially spans both columns, and the text wraps around it.  It seems wrapfig does not work for this in two column, pullquote does not seem to either be exactly what I need.  I've tried using minipage with no luck. I tried to post an example of what I need to accomplish but since I am new this forum I cannot.  I'd be happy to send an image if this helps.   Any suggestions would be fantastically helpful! Maybe this is an easy question, maybe not.  I have a select box where I hardcode with width. Say 120px.    &lt;select style="width: 120px"&gt;   &lt;option&gt;REALLY LONG TEXT, REALLY LONG TEXT, REALLY LONG TEXT&lt;/option&gt;   &lt;option&gt;ABC&lt;/option&gt; &lt;/select&gt;     I want to be able to show the second option so that the user can see the full length of the text.  Like everything else.  This works fine in Firefox, but doesn't work with Internet Explorer6. Whenever we have a changing magnetic flux through a conductor an induced electric field is generated. It is often said that these induced electric field lines are circular in shape.But why is this so?Why are these field lines not elliptical for example? How can this shape be mathematically derived?  Also does this have anything to do with equipotential surfaces?Do such non-conservative electric fields even produce a potential field?If not why? Easy way to turn JavaScript array into comma-separated list? I have a one-dimensional array of strings in JavaScript that I'd like to turn into a comma-separated list. Is there a simple way in garden-variety JavaScript (or jQuery) to turn that into a comma-separated list? (I know how to iterate through the array and build the string myself by concatenation if that's the only way.) How to pluralize an acronym which ends in 'S' correctly? When using the acronym package, how can you  an acronym which ends on 'S'?  Take the acronym: OS - operating system  It should be pluralized as OS's, but is pluralized as OSs when using \acp. Thousands of .log files in Exchange 2007 Server folder I'm a bit of an Exchange admin noob so I don't know if this is normal, but on our Exchange 2007 server under C:\Program Files\Microsoft\Exchange Server\Mailbox\First Storage Group\ there are 30,000 1MB .log files.  I don't know if those should be there or not, but they keep multiplying like little jackrabbits.  I've so far resisted deleting them as I'm sure something would break, but it's come to the point now where our 80GB disk is completely full up and Exchange has stopped receiving messages.  My best guess is that these are transaction logs and would be useful in rebuilding the database should something go awry, but 30GB of transaction logging seems a bit excessive.  Is there a way to limit it or is that even a good idea?  And of course did I completely miss the point? Is there an "exists" function for jQuery? Prove that sequence $ x_{(n+1)}= \frac {a}{1+x_n}$ is convergent to positive root of $x^2+x-a=0,$where $a >0$ and $x_1 >0$ Prove that sequence $$ x_{(n+1)}= \frac {a}{1+x_n}$$ is convergent to positive root of $x^2+x-a=0,$where $a &gt;0$ and $x_1 &gt;0$  we have $$x_{(n+2)}-x_n=\frac{-a(x_{(n+1)}-x_{(n-1)})}{(1+x_{(n-1)})(1+x_{(n+1)})}$$  Hence from this we see behaviour of even terms subsequence and odd terms subsequence  is opposite. how to proceed from there. any hint please Enable thumbnail preview when uploading files in Google Chrome/Firefox? I have google chrome and when I upload photos to Facebook or Imageshack I cannot find a way to show thumbnails, only detail view. I also tried this in Firefox. This is quite important as it is stopping my friend from switching to Ubuntu over Windows.  Included a picture of the File Upload window, how do I  change this to a thumbnail view rather than the current list? I need to calculate the surface (m2) of external walls of all the buildings in a town. By external walls I mean all the walls of a building or parts of a wall which are not in contact with another building. I have a 2D shapefile of polygons with the cadastral footprint of the buildings, which includes information of the building’s height in the attribute table.  I don’t really know how can I do it… My idea is to convert the 2D file to 3D and use the 3D shapefile to distinguish the external walls from the shared walls (parts of a building that are shared with other buildings) of all buildings, and from there, calculate the total area but I’m not sure whether this can be done with ArcScene.  Anybody knows how could it be done and if there is an automate way to do it? I'm new to Ubuntu. I installed it today, but the first thing I encounter is that I can only connect to the Internet through wifi.  Can someone help me?   iwconfig result :  eth0      no wireless extensions.  lo        no wireless extensions.   lspci:  00:00.0 Host bridge: Intel Corporation 3rd Gen Core processor DRAM     Controller (rev 09) 00:01.0 PCI bridge: Intel Corporation Xeon E3-1200 v2/3rd Gen Core processor PCI Express Root Port (rev 09) 00:02.0 VGA compatible controller: Intel Corporation 3rd Gen Core processor Graphics Controller (rev 09) 00:14.0 USB controller: Intel Corporation 7 Series/C210 Series Chipset Family USB xHCI Host Controller (rev 04) 00:16.0 Communication controller: Intel Corporation 7 Series/C210 Series Chipset Family MEI Controller #1 (rev 04) 00:1a.0 USB controller: Intel Corporation 7 Series/C210 Series Chipset Family USB Enhanced Host Controller #2 (rev 04) 00:1b.0 Audio device: Intel Corporation 7 Series/C210 Series Chipset Family High Definition Audio Controller (rev 04) 00:1c.0 PCI bridge: Intel Corporation 7 Series/C210 Series Chipset Family PCI Express Root Port 1 (rev c4) 00:1c.1 PCI bridge: Intel Corporation 7 Series/C210 Series Chipset Family PCI Express Root Port 2 (rev c4) 00:1c.3 PCI bridge: Intel Corporation 7 Series/C210 Series Chipset Family PCI Express Root Port 4 (rev c4) 00:1d.0 USB controller: Intel Corporation 7 Series/C210 Series Chipset Family USB Enhanced Host Controller #1 (rev 04) 00:1f.0 ISA bridge: Intel Corporation HM76 Express Chipset LPC Controller (rev 04) 00:1f.2 SATA controller: Intel Corporation 7 Series Chipset Family 6-port SATA Controller [AHCI mode] (rev 04) 00:1f.3 SMBus: Intel Corporation 7 Series/C210 Series Chipset Family SMBus Controller (rev 04) 01:00.0 3D controller: NVIDIA Corporation GF117M [GeForce 610M/710M/820M / GT 620M/625M/630M/720M] (rev a1) 03:00.0 Network controller: MEDIATEK Corp. MT7630e 802.11bgn Wireless Network Adapter 04:00.0 Unassigned class [ff00]: Realtek Semiconductor Co., Ltd. Device 5289 (rev 01) 04:00.2 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller (rev 0a)   lspci -nnk | grep 0280 -A2:  03:00.0 Network controller [0280]: MEDIATEK Corp. MT7630e 802.11bgn Wireless Network Adapter [14c3:7630]     Subsystem: Foxconn International, Inc. Device [105b:e074] 04:00.0 Unassigned class [ff00]: Realtek Semiconductor Co., Ltd. Device [10ec:5289] (rev 01)