What is the correct way to use dot punctuation to end this sentence? Which of the following way of using . punctuation is correct? There are two sentences here.  Donald Trump became the President of the U.S. The election took place in 2016.    Donald Trump became the President of the U.S.. The election took place in 2016.
What if an abbreviation is at the end of a declarative sentence? Should I duplicate a full stop sign? What if an abbreviation is at the end of a declarative sentence? Should I duplicate a full stop sign? If the following sentence begins with a proper noun, not putting two periods may be confusing, in theory. Edit: It used to be 'contraction' instead of 'abbreviation'.
Proving a sequence defined by a recurrence relation converges How can I prove that this iterative sequence converges to $2$? Can I use the convergence definition?  $$a_{n+1} = \frac{a_n}{2} + \frac{2}{a_n},\qquad  a_0 = 4  $$  Thanks for the help. Why must $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ -- the set of all polynomials in $\sqrt{2}$ and $\sqrt{3}$ with rational coefficients  -- contain multiplicative inverses?   I have gathered that every element of $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ takes form $a + b\sqrt{2} + c\sqrt{3} + d\sqrt{2}\sqrt{3}$ for some $a,b,c,d \in \mathbb{Q}$, and I have shown that $\mathbb{Q}[\sqrt{2}]$ and $\mathbb{Q}[\sqrt{3}]$ are both fields, but it's not clear to me why this allows for general multiplicative inverses to exist in $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$. It seems like to deleted too much files by rm -r and now I have the error of "the system is running a low-graphic mode" in Ubuntu and I have to work in Windows because of that.  However I can load it in recovery mode. I need to copy the files from Downloads folder to a flash drive. Downloads folder has them as I can see by ls command.   When I plug in a flash drive, there will be some messages saying that everything is ok (as I understood) and it's plugged. But when I go to /media/my_name and do ls then I don't see anything. I don't see anything in /run/media/my_name either(I'm not sure if should though).  I can't connect to the Internet for some reason. Ubuntu was installed by wubi.exe file.  How do I save the content of my Downloads folder to a flash drive? Finding $\lim\limits_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$ for continuous $f:[0,1]\to[0,\infty)$ Find $$\lim_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$$if $f:[0,1]\rightarrow(0,\infty)$ is a continuous function.   My attempt:  Say $f(x)$ has a max. value $M$. Then $$\left(\int_0^1(f(x))^ndx\right)^\frac{1}{n}\leq\left(\int_0^1M^ndx\right)^\frac{1}{n} =M$$  I cannot figure out what to do next. I want to make a JavaScript application that's not open source, and thus I wish to learn how to can obfuscate my JS code? Is this possible? Chapterwise page range in the TOC Can you give me some advice for how to obtain the following format of the TOC?        Chapter’s heading 1-7   Chapter’s heading 9-10   Chapter’s heading 11-15 Let $E$ be a metric $\mathbb R$-vector space, $I$ be a nonempty set and $(\mu_t)_{t\in I},(\nu_t)_{t\in I}$ be families of nonnegative finite measures on $\mathcal B(E)$.  that if $(\mu_t)_{t\in I}$ and $(\nu_t)_{t\in I}$ are tight$^1$ and bounded in total variation norm, then the family of convolutions $(\mu_t\ast\nu_t)_{t\in I}$ is tight and bounded in total variation norm as well.  Now I would like to show the converse: If $(\mu_t)_{t\in I}$ and $(\mu_t\ast\nu_t)_{t\in I}$ are tight, then $(\nu_t)_{t\in I}$ is tight. (Most probably we need to impose a boundedness assumption on $(\mu_t)_{t\in I}$ and on $(\mu_t\ast\nu_t)_{t\in I}$ or even a priori on $(\nu_t)_{t\in I}$.)  For the moment, I'm only able to prove the claim assuming that the $\mu_t,\nu_t$ are probability measures. Under this assumption, the proof goes as follows: Let $\varepsilon&gt;0$. Since $(\mu_t)_{t\in I}$ and $(\mu_t\ast\nu_t)_{t\in I}$ are tight, there is a compact $K\subseteq E$ with $$\sup\left(\sup_{t\in I}\mu_t(K^c),\sup_{t\in I}(\mu_t\ast\nu_t)(K^c)\right)&lt;\varepsilon\tag1.$$ Now $K-K$ is compact and \begin{equation}\begin{split}1-\varepsilon&lt;(\mu_t\ast\nu_t)(K)&amp;=\underbrace{\int_K\mu_t({\rm d}x)\nu_t(\underbrace{K-x}_{\subseteq\:K-K})}_{\le\:\nu_t(K-K)}+\underbrace{\int_{K^c}\mu_t({\rm d}x)\underbrace{\nu_t(K-x)}_{\le\:1}}_{&lt;\:\varepsilon}\\&amp;&lt;\nu_t(K-K)+\varepsilon\end{split}\tag2\end{equation} for all $t\in I$, which immediately yields $$\nu_t(K-K)&gt;1-2\varepsilon\;\;\;\text{for all }t\in I\tag3.$$  Can we drop the probability measure assumption?  EDIT I'm not able to obtain the claim, even with the help of the answer below. If $\varepsilon&gt;0$, the goal is to show that $$\nu_t(E)-\nu_t(K-K)=\nu_t((K-K)^c)&lt;\varepsilon\tag4.$$ If I'm choosing all the constants as generic as possible, we obtain $$\mu_t(E)\le c_1\tag5$$ $$\mu_t(K^c)\le k_1(\varepsilon)\tag6$$ $$(\mu_t\ast\nu_t)(K)&lt;c_1\nu_t(K-K)+c_2k_1(\varepsilon)\tag6$$ $$\mu_t(E)\ge\alpha_1\tag7$$ $$(\mu_t\ast\nu_t)(E)\le c_3\tag8$$ $$\nu_t(E)\le\frac{c_3}{\alpha_1}=:c_2\tag9$$ $$(\mu_t\ast\nu_t)(E)\ge\alpha_3\tag{10}$$ $$\nu_t(E)\ge\frac{\alpha_3}{c_1}\tag{11}$$ $$(\mu_t\ast\nu_t)(K^c)&lt;k_3(\varepsilon)$$ The assumptions that I'm imposing is that $(\mu_t)$ and $(\mu_t\ast\nu_t)$ are bounded above (by $c_1$ and $c_3$) and below (by $\alpha_1$ and $\alpha_3$) in total variation norm and are tight (so that we can find $K$ for arbitrary $k_1(\varepsilon)$ and $k_3(\varepsilon)$). In order to obtain the claim we need to choose $k_1(\varepsilon)$ and $k_3(\varepsilon)$ such that $(4)$ holds. From everything above, we obtain $$\nu_t(E)-\nu_t(K-K)&lt;\frac{c_3}{\alpha_1}-\frac{\alpha_3}{c_1}+\frac{c_3}{\alpha_1c_1}k_1(\varepsilon)+\frac1{c_1}k_3.(\varepsilon)\tag{12}$$ Even if we simplify this by assuming $\alpha_1=\alpha_3$, $c_1=c_3$ and $k_1(\varepsilon)=k_3(\varepsilon)$, the equation $$\frac{c_3}{\alpha_1}-\frac{\alpha_3}{c_1}+\frac{c_3}{\alpha_1c_1}k_1(\varepsilon)+\frac1{c_1}k_3.(\varepsilon)=\varepsilon\tag{13}$$ doesn't seem to admit a solution $(k_1(\varepsilon),k_3(\varepsilon))$ with positive $k_1(\varepsilon)$ and $k_3(\varepsilon)$; unless $\frac{c_3}{\alpha_1}-\frac{\alpha_3}{c_1}\ge0$; which clearly doesn't need to be the case (it is clearly the case for probability measures though ...)  $^1$ Remember that a family $\mathcal F$ of finite signed measures on $\mathcal B(E)$ is called tight if for all $\varepsilon&gt;0$, there is a compact $K\subseteq E$ such that $$\sup_{\mu\in\mathcal F}|\mu|(K^c)&lt;\varepsilon.$$ Can I have multiple background images using CSS? Is it possible to have two background images?  For instance, I'd like to have one image repeat across the top (repeat-x), and another repeat across the entire page (repeat), where the one across the entire page is behind the one which repeats across the top.  I've found that I can achieve the desired effect for two background images by setting the background of html and body:  html {     background: url(images/bg.png); }  body {     background: url(images/bgtop.png) repeat-x; }   Is this "good" CSS?  Is there a better method?  And what if I wanted three or more background images? Windows 8.1 missing font files after restart "Items collection must be empty before using ItemsSource." I'm trying to get images to display in a WPF ListView styled like a WrapPanel as described in this old ATC Avalon Team article: .    When I try to populate the ListView with a LINQ-to-Entities queried collection of ADO.NET Entity Framework objects I get the following exception:  Exception     Items collection must be empty before   using ItemsSource.   My code&hellip;  Visual Basic  Private Sub Window1_Loaded(...) Handles MyBase.Loaded     ListViewImages.ItemsSource = From g In db.Graphic _                                  Order By g.DateAdded Ascending _                                  Select g End Sub   XAML  &lt;ListView Name="ListViewImages"           SelectionMode="Single"           ItemsSource="{Binding}"&gt;     &lt;local:ImageView /&gt; &lt;/ListView&gt;   I put a breakpoint on that line.  ListViewImages.ItemsSource is Nothing just before the LINQ assignment. Is it possible to make an EXACT ISO of my system to put on other computers? Vehicles required/allowed in weigh stations on US interstates Show bounded and convex function on $\mathbb R$ is constant Big Endian vs. Little Endian in Qiskit I've noticed that Q# favors Little Endian. Meaning that most operations are designed for this type of encoding.  Is is it the same with Qiskit? Finding Elevation Change, any method So, I think it'd be cool - for goal setting for example - if I could keep track of my elevation change.  The problem is, the local cyclocomputers I looked at seem to use barometric pressure to determine the change.  This is fine, unless some weather is moving in, which happens not infrequently in the Midwestern United States.  Or do some of them try to use an inclinometer, your speed, and some math you learn in Calculus II to keep track of your change?  That would seem more accurate!  I could try to use a GPS meter, but those things are expensive -  and based on back when I had service to my smart phone, not guaranteed to match my car's odometer for distance even - I've read from some other riders that the distance varies at least slightly from local measurement on a bike as well.   Alternatively, let's say that I ride along a known route.  Is there any good way of plotting that route for elevation change?  Thanks! What are the minimum contributions required for co-authorship "Users" folder shared with everyone Today I found something strange on my newly installed Windows 7 OS - my "users" folder C:\Users was shared and "Everyone" had full access... I disabled that and now it seems fine, however I'm a bit scared actually, I'd like to know what caused that and if my system is compromised. How to: Unlimited Bash/shell history? Is there a way to define an unlimited history in Bash ? Chain-calling parent initialisers in python Proving that a doubly-periodic entire function $f$ is constant. Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: $$f(z+\omega_1)=f(z)=f(z+\omega_2), \quad \forall\,z\in \Bbb C.$$Prove that $f$ is constant.   The intuition seems clear to me, we have the three vertices of a triangle given by $0$, $\omega_1$ and $\omega_2$. All points in the plane are one of the vertices of that triangle under a suitable parallel translation. The constant value will be $f(0)$, fine. Throwing values for $z$ there, I have found that $$f(n\omega_1) = f(\omega_1) = f(0)=f(\omega_2) = f(n\omega_2), \quad \forall\, n \in \Bbb Z.$$  I don't know how to improve the above for, say, rationals (at least). Some another ideas would be:   Checking that $f' \equiv 0$. I don't have a clue of how to do that. Write $w = a\omega_1+b\omega_2$, with $a,b \in \Bbb R$, do stuff and conclude that $f(w) = f(0)$. This approach doesn't seem good, because I only have a weak result with integers above. Finding that $f$ coincides with $f(0)$ on a set with an accumulation point. This seems also bad: the set on with $f$ coincides with $f(0)$ by which I found above is discrete.    Nothing works and this is getting annoying... And I don't see how analyticity comes in there. I'll be very thankful if someone can give me an idea.  (On a side note.. I know that this title is not informative at all. Feel free to edit if you come up with something better.)