I tried with defining another function   $h(x):= f(x)-y$ where $ f(\frac{a+b}{2})&lt;y&lt;f(a)=f(b)$  So that $h(\frac{a+b}{2})\cdot h(a)&lt;0$ and $h(\frac{a+b}{2})\cdot h(b)&lt;0$ and  therfore I have two points $x_{1}\in[a,\frac{a+b}{2}] , x_{2}\in [\frac{a+b}{2},b]$ with $h(x_{1})=0=h(x_{2})$.  But I can't tell anything about how much they are apart.
Let $f$ be a real valued and continuous function over $[a,b]$ with $a &lt;b$ such that $f(a)=f(b)$ , My question here is :     Question (claim): How to show that : there  exists  $c \in [a, \frac{a+b}{2} ]$ such that : $f(c)=f(c+\frac{b-a}{2})$.   Note: I have tried intermediate values but I didn't succeed.
What is the relationship between lands and mana in Magic and what do each of them do?  A good answer should especially include how a new player should think about them.    Note:  This is an attempt to create a canonical question about the relationship between lands and mana in Magic.  Beginners are often confused about this given the same symbols are used to mean different things across basic lands, non-basic lands, and mana costs.  This can lead to confusion about , about , or even about . Changing Network Interfaces name Ubuntu 16.04 I tried to change network interfaces name on this new Ubuntu 16.04 LTS version but doesn't have the /etc/udev/rules.d/70-persistent-net.rules.  So, I tried to use /lib/udev/write_net_rules but it not exist.  Why do I need this modification? Because I'm using a tool to Simulate virtual platforms that use flex licensing and the authentication need to be in eth0 interface name.  Any suggestion?  The command ip link returns:  user@laptop:~$ ip link  1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN mode DEFAULT group default qlen 1 link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00 2: enp6s0: &lt;NO-CARRIER,BROADCAST,MULTICAST,UP&gt; mtu 1500 qdisc pfifo_fast state DOWN mode DEFAULT group default qlen 1000 link/ether b8:2a:xx:yy:xx:yy brd ff:ff:ff:ff:ff:ff 3: wlp7s0: &lt;BROADCAST,MULTICAST,UP,LOWER_UP&gt; mtu 1500 qdisc mq state UP mode DORMANT group default qlen 1000 link/ether 5c:e0:xx:yy:xx:yy brd ff:ff:ff:ff:ff:ff  user@laptop:~$ ifconfig  enp6s0    Link encap:Ethernet  HWaddr b8:2a:xx:yy:xx:yy   lo        Link encap:Local Loopback   wlp7s0    Link encap:Ethernet  HWaddr 5c:e0:xx:yy:xx:yy     (Some information were ignored and suppressed)  I already tried some links: Why does no physical energy-momentum tensor exist for the gravitational field? Starting  with the Einstein-Hilbert Lagrangian  $$ L_{EH} = -\frac{1}{2}(R + 2\Lambda)$$  one can formally calculate a gravitational energy-momentum tensor   $$ T_{EH}^{\mu\nu} = -2 \frac{\delta L_{EH}}{\delta g_{\mu\nu}}$$  leading to  $$ T_{EH}^{\mu\nu} = -G_{\mu\nu} + \Lambda g_{\mu\nu} = -(R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R) + \Lambda g_{\mu\nu}. $$  But then, in the paragraph below Eq. (228) on page 62 of  paper, it is said that this quantity is not a physical quantity and that it is well known that for the gravitational field no (physical) energy-momentum tensor exists.  To me personally, this fact is rather surprising than well known. So can somebody explain to me (mathematically and/or "intuitively") why there is no energy-momentum tensor for the gravitational field? I know it is so weird, but last night I set an image on my desktop background and delete the image, then I right-click on the desktop and press the desktop properties to change the style of background (center, stretch and etc) then the desktop properties stopped something like hanging on Windows.   Then a black window alerted something and I press yes without reading it. after that with right-click on the desktop I just see this:    Even some system applications don't work now. for example, I cannot take a screenshot.  Also, my pcmanfm-qt is dead completely, with the command line or clicking on the icon nothing happens.  I don't know what happens. and I have no idea how to fix it. DeleteDuplicates[] does not work as expected on floating point values My opponent casts , which resolves, then he attacks with his creatures.  In my hand, I have a creature card with Flash. If I summon this creature before the declare blockers step, will it be affected by Predatory Rampage's blocking requirement? Get URL of ASP.Net Page in code-behind How to make unit selection circles merge? Listing, zebra effect and broken lines In , I've found one solution to produce a zebra effect for listing. Unfortunately, this only works if the lines are not broken. Is there a way to fix this?  Here is a code with an example showing the problem:  \documentclass[10pt,a4paper]{article}     \usepackage[utf8x]{inputenc}     \usepackage{ucs}     \usepackage[T1]{fontenc}     \usepackage{lmodern}     \usepackage[frenchb]{babel}     \usepackage[x11names, svgnames]{xcolor}     \usepackage{graphicx}     \usepackage{listings}     \usepackage{amsfonts}     \usepackage{amssymb}      \makeatletter         % Patch line number key to call line background macro         \lst@Key{numbers}{none}{%             \def\lst@PlaceNumber{\lst@linebgrd}%             \lstKV@SwitchCases{#1}%             {none&amp;\\%              left&amp;\def\lst@PlaceNumber{\llap{\normalfont                         \lst@numberstyle{\thelstnumber}\kern\lst@numbersep}\lst@linebgrd}\\%              right&amp;\def\lst@PlaceNumber{\rlap{\normalfont                         \kern\linewidth \kern\lst@numbersep                         \lst@numberstyle{\thelstnumber}}\lst@linebgrd}%             }{\PackageError{Listings}{Numbers #1 unknown}\@ehc}}          % New keys         \lst@Key{linebackgroundcolor}{}{%             \def\lst@linebgrdcolor{#1}%         }         \lst@Key{linebackgroundsep}{0pt}{%             \def\lst@linebgrdsep{#1}%         }         \lst@Key{linebackgroundwidth}{\linewidth}{%             \def\lst@linebgrdwidth{#1}%         }         \lst@Key{linebackgroundheight}{\ht\strutbox}{%             \def\lst@linebgrdheight{#1}%         }         \lst@Key{linebackgrounddepth}{\dp\strutbox}{%             \def\lst@linebgrddepth{#1}%         }         \lst@Key{linebackgroundcmd}{\color@block}{%             \def\lst@linebgrdcmd{#1}%         }           % Line Background macro         \newcommand{\lst@linebgrd}{%             \ifx\lst@linebgrdcolor\empty\else             \rlap{%                 \lst@basicstyle                 \color{-.}% By default use the opposite (`-`) of the current color (`.`) as background                 \lst@linebgrdcolor{%                 \kern-\dimexpr\lst@linebgrdsep\relax%                 \lst@linebgrdcmd{\lst@linebgrdwidth}{\lst@linebgrdheight}{\lst@linebgrddepth}%                 }%             }%             \fi         }     \makeatother      \lstset{ %    * Number for lines         numbers=left,                   % Where to put the line-numbers         numberstyle=\footnotesize,      % Size of the fonts used for the line-numbers         stepnumber=1,                   % Step between two numbers         numbersep=5pt,                  % How far the line-numbers are from the code % %    * Back returns for long lines         breaklines=true,                % Sets automatic line breaking         breakatwhitespace=false,        % Automatic breaks only happen at whitespace ?         breakindent=0pt,                % Space before the text of a break         postbreak=\mbox{                % Character used at the begining of a break             \rotatebox[y=0.9ex]{180}{\color{Red} $\Lsh$}%         },     }  \begin{document}  \begin{lstlisting}[language=Python,linebackgroundcolor={\ifodd\value{lstnumber}\color{gray}\fi}] for i in range(5): for i in range(5): for i in range(5): for i in range(5): for i in range(5): for i in range(5):     sum += i**3 print sum  print('Fin de la boucle...') \end{lstlisting}  \end{document} How do I install a Fortran 77 compiler? I have just installed 12.04 LTS on my laptop and I'm kinda new in this. I am required to write some program in Fortran 77 and I need a Fortran 77 compiler. I have tried installing g77 as described in the Ubuntu website but was unable to do so. I have typed in the following in terminal:  sudo apt-get install g77   and get the following:  Reading package lists... Done Building dependency tree        Reading state information... Done Package g77 is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'g77' has no installation candidate   May I know what is the problem? I have tried searching g77 in the software center but apparently it is not there. Does this mean it is not available for 12.04?  I'm currently using an Asus K43S laptop with Intel Core i5-2450M processor and my Ubuntu is installed alongside Windows. Iterating over array of objects javascript - odd behaviour? Centering \vdots in a system of many equations I have a system of equations with an arbitrary number of equations (k). I'd like to use \vdots to compactly describe the system, like so:  \begin{align*}   R(-1) &amp;= \sum_{i=1}^m A(i)R(i-1) \\   R(-2) &amp;= \sum_{i=1}^m A(i)R(i-2) \\         &amp;\vdots                    \\   R(-k) &amp;= \sum_{i=1}^m A(i)R(i-k) \end{align*}     I'd like the dots, however, to be centered with = or the entire equation. Is there an elegant way of centering a column or row with AMSMath? I'm currently using an unholy combination of whitespace operators (\; \, etc.) to get the job done. c stdout print without new line? How to install a "pdf printer"? Is there a way to install a printer that prints to pdf?  There's a lot of discussions about this in the web foruns, but none seens to work. Where can I send feature requests? A coin is tossed three times. The probability of zero heads is 1/8 and the probability of zero tails is 1/8. What is the probability that there is at least one head and at least one tail?  So, if P(zero heads)= 1/8 , then that should be the same of p(all tails)?  We would use the complement rule and Multiplication Rule?   P(at least one head) = 1 - P(no heads) = 1 - 1/8= 7/8 P(at least one tail) = 1 - P(no tails) = 1 - 1/8= 7/8  would I multiply the two values? How can I diagnose and repair missing drive space? Matrix given by $a_{ij} = 1/(i+j)$ is non-singular. How can I get crash data (stack traces at least) from my Android application? At least when working on my own device being retrieved by cable, but ideally from any instance of my application running on the wild so that I can improve it and make it more solid. Can I block incoming calls using wildcard patterns? Recently I have been getting a few unwanted telemarketing calls, all coming from the same source. Each number though is different at the end.  For instance, if the numbers to be blocked are:    123-4567 123-7642 123-8643   It would be convenient for the future to simply block with a wildcard, such as:   123-*   Is this at all possible?