"I touched their forehead[s] with my fingertip."  I'm not certaint whether or not forehead in this example should be singular or plural. There's a group of people, so obviously, multiple foreheads are being touched. However each individual person only has one forehead.
After the therapy, eight children (43%) became able to crawl/move on their back.   Or should I use "on their backs"?   Singular because each child only has one back, or plural because we're dealing with eight backs?
How to find all questions that were duplicates of another question? I am interested to see a list of all questions that were closed because being a dup of another question.  I strongly want to research a particular question and reading all the answers inside the dups will help me a lot.  Is there such feature? if not, this is a feature request I remember reading this book in the late 80's or early 90's. It was a - sort of - semi SciFi contemporary fairy tale.  It was set in Manhattan in contemporary times, near future and (I think) late 1800's. There was a mist wall round New York that ships could disappear into and arrive out at different times.  There was a man with a white horse, the horse could fly but had forgotten how. The man is brought forward from the past but (like the horse) forgets who he is.  An engineer arrives from the future in a gigantic ocean liner and builds a bridge of light to heaven (possibly at the last millennium turn over).   It was a very involved book with a huge cast of characters that all had some part to play in the story.  In the 1800's story there was a girl suffering from consumption, she slept in a glass room on the roof of her house, open to the freezing winter air, it was what kept her alive. She was rich (the family had made money in whaling) but somehow she and The Man (the guy with the white horse, who was pretty much a nobody) met and fell in love (I think).  Some of the story was set around a lake in upstate NY, and one character started off in San Francisco (contemporary times) and traveled to the lake.  Anyone any idea what I'm talking about? Switching a DIV background image with jQuery I am running an application through gdb and I want to set a breakpoint for any time a specific variable is accessed / changed. Is there a good method for doing this? I would also be interested in other ways to monitor a variable in C/C++ to see if/when it changes. When I dip a paper towel in a cup of water the water gets drawn up due to . How is this reconciled with conservation of energy, as it would seem on the surface that the potential energy of the system has gone up with no work being done? Adobe Illustrator: Lines with stroke < 2.0pt doesn't change color I created lines using the line segment tool in Adobe Illustrator (latest Adobe Cloud version) and I can't change the color of the stroke (through the Appearance panel) if the stroke weight is less than 2.0pt.   If the stroke is 2pt or more, then the color around the line changes but not the line itself.    The whole behavior is odd for me as I've never faced in Illustrator before.  Edit: added screen below with the line selected. Why is the decay of a neutral rho meson into two neutral pions forbidden? Should I put myself last? "me and my friends" vs. "my friends and me" or "my friends and I" I've always been taught to put myself last when referring to myself in the same sentence as others but the usage of "me and..." seems to be everywhere these days. The misuse of the word "me" instead of "I" aside, is there some new rule I haven't heard of? Shouldn't we put ourselves last regardless of the "me"/"I" usage?  Examples of "correct" usage:      My friends and I went for some ice   cream. Did you see my friends and me   at the ice cream stand?   Examples of "incorrect" usage:     Me and my friends went for some ice   cream. Did you see me and my friends   at the ice cream stand?   Note: I was also taught that the only person who could put themselves first was the queen. How does Java's serialization work and when it should be used instead of some other persistence technique? I've been lately trying to learn more and generally test Java's serialization for both work and personal projects and I must say that the more I know about it, the less I like it. This may be caused by misinformation though so that's why I'm asking these two things from you all:  1: On byte level, how does serialization know how to match serialized values with some class?  One of my problems right here is that I made a small test with ArrayList containing values "one", "two", "three". After serialization the byte array took 78 bytes which seems awfully lot for such low amount of information(19+3+3+4 bytes). Granted there's bound to be some overhead but this leads to my second question:  2: Can serialization be considered a good method for persisting objects at all? Now obviously if I'd use some homemade XML format the persistence data would be something like this  &lt;object&gt;     &lt;class="java.util.ArrayList"&gt;     &lt;!-- Object array inside Arraylist is called elementData --&gt;     &lt;field name="elementData"&gt;         &lt;value&gt;One&lt;/value&gt;         &lt;value&gt;Two&lt;/value&gt;         &lt;value&gt;Three&lt;/value&gt;     &lt;/field&gt; &lt;/object&gt;   which, like XML in general, is a bit bloated and takes 138 bytes(without whitespaces, that is). The same in JSON could be   {     "java.util.ArrayList": {         "elementData": [             "one",             "two",             "three"         ]     } }   which is 75 bytes so already slightly  smaller than Java's serialization. With these text-based formats it's of course obvious that there has to be a way to represent your basic data as text, numbers or any combination of both.  So to recap, how does serialization work on byte/bit level, when it should be used and when it shouldn't be used and what are real benefits of serialization besides that it comes standard in Java? facing difficulty to create dynamic xpath for City field of UTest website I am beginner in automation field, I am trying to automated one of the module, I have tried many different XPATH to fetch proper XPATH for City field o UTest site. Im trying to automate registration module as one of my assignment. I have tried my different xpath example below:  //xpath="//body/div[@class='pac-container pac-logo']/div/span[2]/span" //div[@class='pac-item']/span[3] //div[@class='pac-item']/span //div[@class='pac-container pac-logo']/div/span[2]/../span[3]   the following code written:  //xpath to send data in field @FindBy(xpath="//input[@placeholder='Enter a location']") WebElement city;  //xpath for list to appear @FindBy(xpath="div[@class='pac-item']/span") List&lt;WebElement&gt; City;  public void EnterSignupTwo(String CiTy) throws Exception {     city.clear();     city.sendKeys(CiTy);     Thread.sleep(3000);     System.out.println("the size of List " +City.size());//printing list size      for(int i=0; i&lt;City.size(); i++ ) {         String cities = City.get(i).getText(); //storing list element in String         System.out.println("Converted value of city :"+cities); //print of total list      if(cities.contains(CiTy)) //if parameterized match with Available then click     {         System.out.println("enter specific value of city :"+City.get(i).getText());         City.get(i).click();         break;     } }   kindly refer the $6^{(n+2)} + 7^{(2n+1)}$ is divisible by $43$ for $n \ge 1$ How should I structure my URLs for both SEO and localization? Can we have a perfect field of characteristic $p$ with a non-archimedean valuation? How do I change my update server in Ubuntu 18.04? How to install HPLIP on my Ubuntu to support my HP printer and/or scanner? Let's assume that one has some HP printer and/or scanner and checked its support status on the corresponding page of HPLIP site - .  So one have the following information: the device Model name, Min. HPLIP Version and Driver Plug-in need.   What should one install to make Ubuntu fully support the HP printer and/or scanner? Can one of Newton's Laws of motion be derived from other Newton's Laws of motion? Can one derive Newton's  second and third laws from the first law or   first and third laws from the second law or  first and second  laws from the third  law  I think Newton's laws of motions are independent to each other. They can not be derived from one another. Please share the idea. Baby Rudin claim: $1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\frac{1}{9}+\frac{1}{11}-\frac{1}{6}...$ converges This sequence is a rearrangement of the series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}...$. Note that at this point in the text we do not have any theorem about the convergence of rearrangements.  Let $\{s_n\}$ be the sequence of partials sums of the series then for $n \ge 0$  $s_{3(n+1)} = \sum ^n _ {k=0} \frac{1}{4k+1} + \frac{1}{4k+3} - \frac{2}{4k+4}$  We can view it as the sequence(on $n$) of partials sums of  $\sum_0 a_n = \sum_0 \frac{1}{4n+1} + \frac{1}{4n+3} - \frac{2}{4n+4}$  Where $|a_n| = a_n = \frac{1}{4n+4}\{\frac{3}{4n+1}+\frac{1}{4n+3}\} \le \frac{1}{4n^2}$.  By the comparison test $s_{3(n+1)}$ converges to some real $\alpha$.  But $s_{3(n+1)+1} = s_{3(n+1)}+ \frac{1}{4n+5}$ and $s_{3(n+1)+2} = s_{3(n+1)}+ \frac{1}{4n+5}+\frac{1}{4n+7} $hence we have a partition of $\{s_n\}$ into subsequences which tend to $\alpha$ and this implies $s_n \rightarrow \alpha$.  Is my proof correct? Any alternative solutions are appreciated. Formal power series ring, norm. Let $k$ be a field. Let $R$ be the formal power series ring $k[[x]]$. Define $N$ on $R \setminus \{0\}$ as follows: $N(f)$ is the smallest $n$ of which the coefficient of $x^n$ in $f$ is nonzero.  (a) Prove that $R$ is a Euclidean domain with Euclidean norm $N$.  (b) For $a$, $b$, $a+b$ nonzero elements of $R$, prove that $N(a+b)$ cannot be bounded as a function of $N(a)$ and $N(b)$.  (c) Prove that if $a$ and $b$ are two power series such that $b$ does not divide $a$ (and $b \neq 0$), there are infinitely many pairs $(q, r)$ for which $a = bq + r$ and $N(r) &lt; N(b)$. Prove that for any $x_0 \in X$ and any $r&gt;0$, the open ball $B_r(x_o)$ is open.   My attempt: Let $y\in B_r(x_0)$. By definition, $d(y,x_0)&lt;r$. I want to show there exists an $r_1\in\mathbb{R^+}$ s.t. $B_{r_{1}}(y)\subseteq B_r(x_0)$. Let $a\in B_{r_{1}}(y)$. Then, $d(a,y)&lt;r_1$. For $a\in B_{r}(x_0)$, $d(a,x_0)&lt;r$. I want to show $d(a,y)&lt;r_1$ implies $d(a,x_0)&lt;r$. By triangle inequality, $d(a,y)\leq d(a,x_0) + d(y,x_0) \rightarrow$  $d(a,y)&lt;r_1\leq d(a,x_0)+d(y,x_0)&lt;2r...$  I'm a little stuck after this point. How do I get my PCs to not be a bunch of murderous cretins? Most RPGs teach you that casual violence is the best solution to all your in-game problems. This is so well established a part of the vast majority of RPGs that there are entire satire RPGs like Greg Costikyan's  and John Tynes'  dedicated to showcasing the issue. Even now that the '80's anti-RPG hysteria is past, you see more sober critiques of endemic violence in RPGs in places like . In most RPGs, PCs become inured to murder and other antisocial activities very quickly, and enter the depths of depravity that wouldn't be appropriate in the worst parts of the world. Armed robbery, mass murder, and genocide become routine parts of an adventurer's day, something only stick-in-the-mud characters with the most extremely stated ethics object to. , even though it is not properly applied to just any conflict, is a PC's friend and they generally escalate any conflict in that direction. The sophistication of the gamer mindset towards introspection on this issue can be demonstrated in that the most meaningful question usually debated about in-game violence is some variant on &quot;but should we kill the noncombatant children&quot; or &quot;can we just murder people out of hand as long as they're from a typically evil race?&quot; In the real world, we generally a priori regard anyone having to have this discussion as a monster already. (P.S. The &quot;wipe out a given evil race&quot; thing isn't the point of this question; it is only mentioned to indicate that it's only &quot;that far&quot; that usually causes ethical handwringing from players, when that is a really quite extreme case and we should be uncomfortable with casual violence a lot sooner. ) What I'd like to have is a more realistic in-game treatment of conflict.  People getting wounded and giving up, taking and ransoming of prisoners, not always escalating a fistfight to weapons, not always escalating weapon combat to killing, etc.  Heck, as I write this, I'm watching an episode of Adventure Time with my daughter and the protagonists snuck in and rescued a princess kidnapped by the Ice King and then exit right next to his sleeping form. I thought, &quot;If this were D&amp;D they'd all be carefully coordinating a coup de grace to kill him in his sleep on the grounds that he inconvenienced them.&quot;  The problem isn't limited to D&amp;D of course, sci-fi PCs are happy to neutron bomb planets for convenience too, for example. I'm not part of the &quot;D&amp;D is Satanic&quot; crowd obviously, but I frankly do have compunctions about continually playing in games where the taught behavior is uncomfortably equivalent to the worst examples of human behavior we see on the nightly news. How can I give my PCs a newfound respect for human life?