Compute the fresnel integral $\int_0^x\cos(\frac{\pi u^2}{2})$du
Asymptotic expansion for Fresnel Integrals
I keep going around in circles with this problem:  Given the functions $f:X\to Y$ and $g:Y\to Z$, if $g\circ f:X\to Z$ is bijective, then prove $f$ must be one-to-one and $g$ must be onto. I was able to prove this without difficulty. However the second part of the problem has me stumped:  a) Find functions $f,g$ where $g\circ f$ is bijective but $f$ fails to be onto.  b) Find functions $f,g$ where $g\circ f$ is bijective but $g$ fails to be one-to-one.  My attempt for (a): Let $f:\mathbb{R}\to \mathbb{R}^+$ be defined by $f(x)=e^x$ and $g:\mathbb{R}^+\to \mathbb{R}$ be defined by $g(x)=\ln(x)$. Then $g\circ f(x)=g(f(x))=g(e^x)=\ln(e^x)=x$ which is bijective, $g$ is onto and $f$ fails to be onto... or so I thought. Upon closer inspection $f$ $\underline{is}$ onto for the given codomain $\mathbb{R}^+$.  And now I'm stuck in a rut. Anyone out there have any hints? Proof that the irrational numbers are uncountable I created the shiftcommandn shortcut for Firefox in Syst Prefs to match other browsers' behavior.     It registers in the Firefox Menubar.     But pressing the shortcut opens a new non-private window, and it loads there the first item in my "Top Sites".   I imagine I could create another shortcut to override this behavior, but what is this behavior? I don't see anything highlighted in the Firefox Menubar when I hit the shortcut. Or might there be another solution (other than using a different shortcut to open new private windows)?   Problem exists on both Mojave and High Sierra. Default Firefox shortcut (shiftcommandp) still opens a new private window. Layover at Dusseldorf and Frankfurt, category A or B visa? I am a Indian citizen in the United States on a F-1 visa.  For the summer break, I am traveling (round-trip) from Chicago to New Delhi via Dusseldorf-Frankfurt by Lufthansa airlines. My onward Journey would be on July - 8 and return Journey on Aug - 11.  My layover will be of 1 hour 10 mins in Frankfurt and 3 hours in Dusseldorf.  I would like to know if I require Category A or Category B Visa. If not, what type of visa do I require for the Schengen visa for my transit. How I can apply for the same?  I tried to find slots in German consulate but all slots are full before my departure date. If I don't get the Schengen visa, I cannot travel and will loose all the fare (since my ticket is a non-refundable ticket). Is there any other way I can tackle this situation. Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional? It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this? Here I offer a "proof" of why the power set of the natural numbers is countable by creating a supposed bijection. I know that my proof is wrong, but where? If anyone could point it out, that'd be awesome.  Consider the power set of the natural numbers. We can form a bijection $f:\mathbb{N}\to\mathcal{P}(\mathbb{N})$ as follows.  For any $f(x)$ convert $x$ to its binary representation, which is a string of $0$s and $1$s. If the $i$-th digit is $1$, then include $i$ in the output set $S=f(x)$. Otherwise do not include $i$. Note that $i$ is $0$-indexed and starts from the right-most digit.  So in this case $f(5)=\{1,3\}$. Supposedly we can construct any set in $\mathcal{P}(\mathbb{N})$ with an appropriate choice of $x$.  Where does this proof blow up? Additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear. A function $T:V \rightarrow W$ is additive if $T(x+y) = T(x) + T(y)$ for every $x, y \in V$. Prove that there exists an additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear.  My attempt: Let $T$ be the function $T: \mathbb{R}$ (over the field $\mathbb{Q}$) $\rightarrow \mathbb{R}$ (over the field $\mathbb{R}$). The set $\{1, \sqrt{2}\} \subseteq \mathbb{R}$ is linearly independent for the vector space $\mathbb{R}$ over $\mathbb{Q}$. Then, there must exist a linearly independent set $W \subseteq \mathbb{R}$ (over $\mathbb{Q}$) such that $\{1, \sqrt{2}\} \subseteq W \subseteq \text{span}(W) = \mathbb{R}$ (over $\mathbb{Q}$).  I have been told that the function defined as $T(1) = 1$ and $T(w) = 0$ for all $w \in \text{span}(W) \setminus \{1\}$ is additive but not linear, but I cannot see why this is? I can see why it is not linear, clearly $T(\sqrt{2} \cdot 1) = 0$ but $\sqrt{2} T(1) = \sqrt{2}$. But, why is $T$ additive? For example, $T(1+1) = T(2) =0$ but $T(1) + T(1) = 1+1 = 2$? Is there a mistake somewhere? Why is the observable universe so big? Could someone advise me on how to determine all entire functions $f$ such that $|f'(z)| \leq |f(z)|, \forall z\ ?$  Hints will suffice, thank you. Duplicate id in list My service is working fine for single record insert when i send bulk records at that time it showing soapenv:ClientSystem.ListException: Duplicate id in list: 01ti0000002gUP1AAM Class.Sample.behaviourOfWebService: line 235, column 1  the error line is Database.upsert(pList)     p.productCode = pCode;    p.name =pName;    pList.add(p);  } How to force the browser to reload cached CSS and JavaScript files Let $A=\begin{pmatrix}0&amp;5\\ 7&amp;4\end{pmatrix}$.      1) Write $A$ as a product of $4$ elementary matrices.      2) Write $A^{-1}$ as a product of $4$ elementary matrices.   My work. I have managed to find $A^{-1}$, which came out to be this:  \begin{pmatrix}-\frac{4}{35}&amp;\frac{1}{7}\\ \frac{1}{5}&amp;0\end{pmatrix}  However, I am struggling to figure out how I would split each of these matrices into $4$ elementary matrices. Any help? HTML table headers always visible at top of window when viewing a large table "Much of it is" or "Much of it are"? I'm writing an college-entrance essay and for me it's imperative that I understand and implement the correct one of these two expressions:     "Much of it is" or "Much of it are"   If you need context:     I'm having trouble with my writing. The reason I'm having trouble is that much of it is about....   Should there be an "is" or an "are"? I just finished installing mysql-server, and then installed phpmyadmin which came with Apache and PHP. It set it up alright and made a phpMyAdmin database, but I can't seem to find the URL to get to it -- I tried going to  but it doesn't seem to be there.  I can't find the location in any of the manuals, does anyone know where this is located? Thanks. UK visit visa: What documents prove genuine relationship with the visitor? How to simulate a rope? I'm trying to simulate a rope (that later might get pulled on the ends*) that interacts with some obstacles. So far I created a plane as the floor and a cube as an obstacle, and activated "collision" in the physics menu for both of them.   *(In the end the goal is simulating a .)  For modelling the rope, I tried using a long cylinder with enough subdivisions, and activate "cloth" in the physics menu. This resulted in the cylinder collapsing like a sleeve of a sweater, but I would like the cross section to remain roughly circular. Is there a way to directly avoid this?   Another approach I tried now was just using a a line made up of vertices and edges (with the idea of later adding modifiers to give it a thickness), but this just fell through all the collision objects as it has no faces. Is there good way to do this? How do I adjust the Clock Indicator format? Why use static_cast(x) instead of (int)x? I've heard that the static_cast function should be preferred to C-style or simple function-style casting.  Is this true?  Why? Note: This is an attempt to create a canonical question that covers all instances of &quot;low-graphics mode&quot; error that occurs to a user, including but not limited to installation of wrong drivers, incorrect or invalid lightdm greeters, low disk space, incorrect installation of graphics card like ATI and Nvidia, incorrect configuration of xorg.conf file while setting up multiple monitors among others. If you are experiencing the &quot;low-graphics mode&quot; error when trying to login but none of the following answers work for you, please do ask a new question and then update the answers of this canonical question as and when your new question gets answered.   When I try to boot into my computer, I am getting this error:  The system is running in low-graphics mode Your screen, graphics cards, and input device settings could not be detected correctly. You will need to configure these yourself.   How do I fix the failsafe X mode and login into my computer?  Answer index: