Let $f,g$ be Riemann integrable on $[0,1]$ such that $\int_0^1 f=\int_0^1 g=1$. Show that there exists $0\leq a&lt;b\leq 1$ such that $\int_a^b f=\int_a^b g=\frac{1}{2}$.  If there is only one function $f$, it is easy. What about two functions?
Let $f(x)$ and $g(x)$ be two continuous functions on $[0,1]$ and  $$\int_{0}^{1}f(x) dx= \int_{0}^{1}g(x)dx = 1$$   Show that there exist  $[a,b]\subset [0,1]$, such that  $$\int_{a}^{b}f(x) dx= \int_{a}^{b}g(x)dx = \frac{1}{2} $$  The question can be solved by considering the fundamental group of $S^1$, now I am wondering if we can solve it by real-analysis.  Here is the topological solution:  Assume that for any $[a,b]\subset[0,1]$, we always have  $$(\int_{a}^{b}f(x) dx\neq \frac{1}{2}) \quad \vee \quad(\int_{a}^{b}g(x)dx \neq \frac{1}{2}) $$ Consider mapping $$\phi:D\rightarrow\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\},\,\,(x,y)\mapsto(\int_{y}^{x}f(t) dt,\int_{y}^{x}g(t) dt)$$ where $D=\{(x,y)|0\leq x\leq y\leq 1\}$.  Let $a$ be path from $(0,0)$ to $(0,1) $in $D$ and $b$ be path from $(0,1)$ to $(1,1) $in $D$, then $ab$ is a path from $(0,0)$ to $(1,1) $in $D$ and $$\phi\circ (ab):[0,1]\rightarrow\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\} $$ Notice that $$(\phi\circ (ab))(0)=(\phi\circ (ab))(1)=(0,0)$$  so $\phi\circ (ab)$ is a loop based on $(0,0)$ in $\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\}$.  For any $t\in [0,\frac{1}{2}]$, it's not hard to get that $$(\phi\circ (ab))(t+\frac{1}{2})=(1,1)-(\phi\circ (ab))(t)$$ is equivalent to $$(\phi\circ (ab))(t+\frac{1}{2})-(\frac{1}{2},\frac{1}{2})=-((\phi\circ (ab))(t)-(\frac{1}{2},\frac{1}{2}))$$  Define retraction  $$r:\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\}\rightarrow S^1,\quad (x,y)\rightarrow\ \frac{(x,y)-(\frac{1}{2},\frac{1}{2})}{||(x,y)-(\frac{1}{2},\frac{1}{2})||}$$  Then $r\circ \phi\circ (ab)$ is a loop on $S^1$, such that $$(r\circ \phi\circ (ab))(t+\frac{1}{2})=-(r\circ \phi\circ (ab))(t),\quad\forall t\in [0,\frac{1}{2}]$$  So $&lt;r\circ \phi\circ (ab)&gt;$ is not trivial in $\pi_1(S^1)$.   However, $ab$ is path homotopic to $c$ in $D$, where $c$ is the path between $(0,0)$ and $(1,1)$ in $D$. In this case, $r\circ \phi\circ (ab)$ is a point-path in $S^1$ by $$(\phi\circ c)(t)=\phi(t,t)\equiv (0,0)$$  which leads to contradiction.
How can I echo a newline in a batch file? How can you you insert a newline from your batch file output?  I want to do something like:  echo hello\nworld   Which would output:  hello world Where does gravity get its energy from? What counts as a "visual illusion" for a character that has Truesight? Creatures with Truesight, have a number of benefits, one of which is that they     automatically detect visual illusions and succeed on saving throws   against them   How do you determine if a spell creates a visual illusion? Is it as simple as "If the illusion creates something that the target can see, it's visual" or is there a more specific rule?  For some context, this came up into play in our last session where a Wizard PC cast Mental Prison on an enemy with Truesight an we weren't exactly sure how to rule it at the time and decided to research it after the session was over. if $x+2y+z=4$ and $x,y,z$ are real number. then find maximum value of $xy+yz+zx$  putting $x+z=4-2z$ in $y(x+z)+zx = y(4-2z)+zx = 4y-2yz+zx$  i wan,t be able to  go further,could some help me with this postfix - how to prevent users from sending with another users name Even though my postfix server cannot be used without authentification (open relay), sending mail in another users name after logging in is still possible like this:  EHLO domain.org auth plain eW91IGxpdHRsZSBkaXJ0eSBiYXN0YXJkIDstKQ== 235 2.7.0 Authentication successful mail from: otheruser@domain.org 250 2.1.0 Ok rcpt to: recipient@otherdomain.org 250 2.1.5 Ok data 354 End data with &lt;CR&gt;&lt;LF&gt;.&lt;CR&gt;&lt;LF&gt; Please send me your account info. . 250 2.0.0 Ok: queued as D40692A61AA quit   How can this be prevented? Resizing an Image without losing any quality What is the word for act of monotonously demanding for something (many be a toy) repeatedly? How do I find out what fonts are used in a document/picture? I know that the gravitational interaction of antimatter is expected to be the same as normal matter.  But my question is, has it ever been experimentally validated?  I think it would not be a trivial experiment, because electromagnetic effects have to be eliminated, so neutral particles would be needed. Maybe diamagnetically trapped antihidrogen atoms could be examined as to which direction they fall? How To Get Notified When Hard Drives Fail In Dell PowerEdge Servers I have a Dell PowerEdge R415 and R515 and wondering how I can be alerted remotely when a hard drive fails in either server? Both are running RAID, so I need to be notified, so I can swap out the drive. Both servers only have the basic Baseboard Management Controller. Is this possible with a third party tool/software? I have looked it up and the resources all talk about schengen visa, travelling with eu national family member etc. so I would like to get a more assuring answer about my situation:   I am a non-EU national with an Italian residence permit. My father is an Italian citizen. I will be travelling to Sofia and Bucharest next month with my girlfriend who is a British citizen. Do I need to get a visa to enter these two cities? After seeing a question regarding the acquisition of Data's cat Spot, the point was brought up that Data's  midway through the series.  My question is, how? I have thought up two possibilities (see below to my own answer) but I'm wondering if there aren't possibly others that might fit better. Exhibit A- Spot the Younger  Exhibit B- Spot the Older Linearlayout order is reversed on some devices Fermi's golden rule and infinite probablity? I am slightly confused about the application of . Which during standard derivations indicates a probability of transitioning from the state $|i \rangle$ to the state $|f\rangle$ of: $$P=\frac{2\pi t}{\hbar^2} | \langle f | \hat V | i \rangle|^2 \delta(E_f-E_i-\hbar \omega)$$ This holds for large values of $t$ but for arbitrarily large values of $t$: $$P&gt;1$$ Which (to me anyway) makes no physical sense since there can't be a probability greater then 1 of been in the final state. If my interpretation is right then how is this problem resolved/allowable? and if it is wrong what part of what I have said is wrong? I think we need a button that says Edit and Close added to the Review Close button options.  Many of these closed posts will be around forever (and should be, closed should not just be a path to deletion), so they need editing.  If I edit it now it goes away, with the only option being a back click to get back to the review, and then click into the post and edit it.  We have an Edit and Reopen button, so this would fit in just fine. Varnish cache with Nginx SSL termination How can I use varnish cache with nginx SSL termination for my SSL-Only (Https only) website? I use ubuntu 18.04 and I am currently running a wordpress site on my server. Only squares instead of letters and numbers are displayed in my calculator and other apps in Ubuntu 18.04 I don't know what I did wrong in my Ubuntu 18.04 OS. For some reason some of my apps look like this:    This also happens to other gnome apps like pdf-reviewer and Logs. Window title, menu bar etc. are all blank, display only squares.  *Problem solved: I downloaded Google Fonts, saved them in a new folder, they can be applied system-wide, but after I did some font settings using Google Fonts it didn't work. I changed the font of Interface with Tweaks-Tool, that was what causing the problem. I almost always want the profile correct applied, but it's an extra step. Any ideas on how to apply these automatically? What's the best counter strategy against battlecruisers? How should I prepare my army when attacking someone who has 12+ battlecruisers? (Protoss/Zerg/Terran answers are more than appreciated) A folder was created in my home folder (not by anyone), and I cannot see the folder contents. She has root privileges, but when the nautilus root access the folder is empty. When I try to remove it, it will appear to remove more than 1 million files from my computer, and I do not know what these files are.  How i can see these files? the name of folder created is "4F8ZBREShQ". I'm scared about it...