I teach children aged 10 to 14. I teach children ages 10 to 14.
Here is a sentence excerpted from ,     Teen suicide is a growing health concern. It is the third-leading cause of death for young people ages 15 to 24, surpassed only by homicide and accidents, according to the U.S. Center for Disease Control and Prevention.   I've heard people use both "people aged from X to Y" and "people ages from X to Y" quite often, but I've always been believing that only the former is grammatically correct, whereas in the latter the "ages" doesn't seem to serve as any grammatical component of this sentence. So when I see an APA paper use "ages" in that way, I am just quite surprised: I just can't believe such an authoritative and rigourous academic institution will let such an explicit (and indeed naive, even to a non-native speaker's eyes) grammatical mistake survive in their publications. I'm aware that in daily language usage one ought not to become so fussy over insignificant matters like this, but I do wonder whether it should be paid attention to in academic writings.  OR, using "ages" in that way is no longer considered a grammatical mistake? Honestly I don't think this'd be true, but if it really is, I'd be grateful to be informed. Thanks!
Sum of reciprocals of numbers with certain terms omitted I know that the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges. I also know that the sum of the inverse of prime numbers $\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \cdots$ diverges too, even if really slowly since it's $O(\log \log n)$.  But I think I read that if we consider the numbers whose decimal representation does not have a certain digit (say, 7) and sum the inverse of these numbers, the sum is finite (usually between 19 and 20, it depends from the missing digit). Does anybody know the result, and some way to prove that the sum is finite? Maximum and minimum of $f(x)=\cos(\sin(x))-\sin(\cos(x))$ Given the function: $$f(x)=\cos(\sin(x))-\sin(\cos(x))$$ it has absolute maxima at $x=(2k+1)\pi$ with $k=0,1,..N$ and relative maxima at $x=2k\pi$. It is not clear where are the minima. Putting the derivative to zero doesn't help. Any suggestion on how to find the minimum value and where is it? Thanks. In the Harry Potter world, there's the term "muggle" for people with no magical abilities, "muggle-born" for those with magical abilities who's parents are muggles and "squib" for those with no magical abilities who's parents are magical.  But is there any collective term for all magical people (as opposed to wizards for males and witches for females)?  I don't recall seeing any word that refers to all men and women and children with magical abilities, and it's the only group left without noun or collective noun to describe them. What is IEEE 802.3 Ethernet's communication type: Synchronous or Asynchronous? In the IEEE 802.3 frame structure there is a 7 byte preamble section for synchronization. If it is really for synchronization does that mean Ethernet is synchronous and not asynchronous?    Ethernet structure for reference: Some days I'm working with extremely large Excel files.  They have too many sheets that can't be shown in one sheet tab. (see image)      So, I'm wondering whether or not we can show Excel sheet on more than 1 row. Many tries with Google don't help much.    Many tries but I still can't make Excel show the sheet bar the way I want (≥ 2 rows), so I'm working with some VBA Scripts kopischke suggested.    FYI, right click on |&lt; &lt;&lt; &gt;&gt; &gt;| button on the bottom-right corner show you a list of up to 15 sheets.  For more, click on View more in the list appeared. What's the intuition behind the identities $\cos(z)= \cosh(iz)$ and $\sin(z)=-i\sinh(iz)$? I'm trying to understand in an intuitive manner the relationship between the circular and hyperbolic functions in the complex plane, i.e.:  $$\cos(z)= \cosh(iz)$$ $$\sin(z)=-i\sinh(iz)$$  where $z$ is a complex number.  From a geometric point of view, what I understand is that cos is the composition of a rotation through $\frac{\pi}{2}$, followed by cosh, and sin is the composition of a rotation through $\frac{\pi}{2}$, followed by sinh, followed by a rotation through $-\frac{\pi}{2}$ (where sin, cos, sinh, cosh are defined as complex functions).  Where does this connection come from? Is there some way it can be visualized in terms of complex mappings? (I'm not asking for a proof of the identities, I already know one). Using a keyboard shortcut to paste text Integral of $\sin(e^t)dt$ Let $f(x) = \int_x^{x+1} \sin(e^t)dt$. Show that $$ e^x|f(x)| &lt; 2 $$ and that $$ e^xf(x) = \cos(e^x) - e^{-1}\cos(e^{x+1}) + r(x) $$ where $|r(x)| &lt; Ce^{-x}$ for some constant $C$.  I can do the first part easily. Integration by parts gives us $$ f(x) = \frac{\cos e^x}{e^x} - \frac{\cos e^{x+1}}{e^{x+1}} - \int_{e^x}^{e^{x+1}} \frac{\cos u}{u^2}du, $$ so substituting $\cos u = 1$ and $\cos u = -1$ gives \begin{align*} \cos e^x - 1 + \frac{1}{e}\left(1- \cos e^{x+1}\right) &amp;\leq e^xf(x) \leq \cos e^x + 1 - \frac{1}{e}\left(\cos e^{x+1} + 1\right)\\ -2 &amp;&lt; e^xf(x) &lt; 2 \end{align*} So that $e^x|f(x)| &lt; 2$.  I get stuck on the second part, though. From integration by parts, I have $e^xf(x) = \cos e^x - e^{-1}\cos e^{x+1} - e^x\int_{e^x}^{e^{x+1}} \frac{\cos u}{u^2}du$, so it reduces to showing that $$ \left|\int_{e^x}^{e^{x+1}} \frac{\cos u}{u^2}du\right| &lt; Ce^{-2x} $$ for some constant $C$. However, I always end up with a constant for $Ce^{x}$ rather than for $C$. I have a JavaScript object like the following:  var p = {     "p1": "value1",     "p2": "value2",     "p3": "value3" };   Now I want to loop through all p elements (p1, p2, p3...) And get their keys and values. How can I do that?  I can modify the JavaScript object if necessary. My ultimate goal is to loop through some key value pairs and if possible I want to avoid using eval. Is it possible to print to the Report window in the Info view? Do professors get paid for supervising students? Preparing CSV files for use in ArcGIS Desktop? When can you switch the order of limits? Top of answer covered by sticky top bar after clicking direct link With the introduction of the network-wide sticky top bar , I noticed a bug that's happening on every site (except for Stack Overflow). When you click on an answer link (of the form https://site.stackexchange.com/a/####), the very top of the answer and the upvote button are covered by the top bar:    Stack Overflow shows the correct behavior: Evaluate $\lim_{n\rightarrow\infty} \int_0^1 f(x^n) dx$ Let $f$ be a real, continuous function defined for all $0\leq x\leq 1$ such that $f(0)=1$, $f(1/2)=2$, and $f(1)=3$. Show that  $$\lim_{n\rightarrow\infty} \int_0^1 f(x^n) dx$$  exists and compute the limit.  Attempt:   Since $f$ is real-valued continuous on $0\leq x\leq 1$ and the boundaries $f(0)=1$ and $f(1)=3$, $f$ is bounded on the interval  and the integral $\displaystyle\int_0^1 f(x^n)dx$ exists for any positive $n$. Thus, we can interchange the limit and the integral and compose the limit, \begin{align*} \lim_{n\rightarrow\infty} \int_0^1 f(x^n)dx&amp;=\int_0^1\lim_{n\rightarrow\infty}f(x^n)dx\\ &amp;=\int_0^1 f\left(\lim_{n\rightarrow\infty} x^n \right)dx\\ &amp;=\int_0^1 f(0)dx=1. \end{align*}  Questions: So, my first question concerns whether interchanging the limit and the integral is correct. It seems it would be justified by the dominated convergence theorem, where $f(x^n)$ is dominated by a function $g(x)=\displaystyle\sup_{0\leq x\leq 1} f(x^n)$ and $f_n(x^n)$, on $0\leq x\leq 1$, converges pointwise to a function $f$ that takes value 1 for $x\neq 1$ and 3 for $x=1$.  My next question is whether the interchange between the limit and composition of the function is allowed. Since the function is continuous and the limit exists at that point, it seems the interchange would be justified. How to prove one has spent time in the non-Schengen European microstates, in order to pause the Schengen clock? I am considering staying in Monaco, San Marino, or Vatican City before I can enter the Schengen area again. , it is valid to stay in these three micro-states in order to stop the Schengen clock. However, also according to the answer:  Problems will arise when the visitor ultimately leaves the zone and must undergo a Schengen exit inspection. At this point the person may be asked to prove that they are not an overstayer and the border guard will be entitled to see hard evidence in the form of official documentation. There was a case where an individual attempted to show that his mobile phone registered to a tower outside of the zone and this failed obviously: radio waves have no respect for nation borders but also one could mail the SIM card to a friend in a far away land, put in a phone where it registers to the tower and then mail it back. Apparently hotel receipts can be too easily forged to be successful as evidence. What the border guard will want to see is a passport stamp from the local constabulary. But the constabulary is under no obligation to issue one, so success is down to personal impact and articulation skills.  I'm not sure if I could get a stamp from the local constabulary. However, I always use AirBnB to stay, and the reservation email and the mobile app might be far more reliable than a random hotel reservation document. So, does the AirBnB reservation email work as a proof of staying there? Or is there any better document as a proof of staying (&quot;better&quot;, in terms of ensuring that I don't pay the overstaying penalty)? I will likely leave the Schengen area ultimately from Frankfurt International Airport. I am failing to add a heading for the appendix to the ToC with the correct page number:  \documentclass{scrbook} \begin{document} \tableofcontents \chapter{test} \appendix \addcontentsline{toc}{part}{\appendixname} \chapter{First Appendix} \end{document}     I also tried to use the \addappheadtotoc command from the  package, but that changes nothing (expect that it writes 'Appendices' instead).  I assume that the appendix package provides the solution, but I could not figure out how to use it. Why are there only derivatives to the first order in the Lagrangian? Find four groups of order 20 not isomorphic to each other and prove why they aren't isomorphic.   So far I thought of $\mathbb Z_{20}$, $\mathbb Z_2 \oplus\mathbb Z_{10}$, and $D_{10}$ (dihedral group), but I can't find another one. Would $U(50)$ work? I know it has order 20 and is cyclic but I'm not exactly sure how to move from there. Can someone to point me on the right direction? Inequality involving sides of a triangle How can one show that for triangles of sides $a,b,c$ that   $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} &lt; 2$$  My proof is long winded, which is why I am posting the problem here.   Step 1: let $a=x+y$, $b=y+z$, $c=x+z$, and let $x+y+z=1$ to get   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}&lt;2$  Step 2: consider the function $f(x)=\frac{1-x}{1+x}$, and note that it is convex on the interval (0,1), so the minimum of   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}$  is reached when the function takes the extreme points. i.e. $x=y=0, z=1$.    But going back to the fact that this is a triangle, we note that $x=y=0 \implies a=0$ which is not possible, so the inequality is strict.