(..) those who stand to profit from our working harder,   I've noticed this pattern very often, but I don't know how or when to use it. The following      (...) those who stand to profit from us working harder,   would be right as well, right? Is there a difference in meaning? Or in context of use maybe? Thanks in advance!  For reference, the sentence is taken from the fourth paragraph from the end in .
I assume that the following sentences are grammatically correct:   He resents your being more popular than he is. Most of the members paid their dues without my asking them. They objected to the youngest girl's being given the command position. What do you think about his buying such an expensive car? We were all sorry about Jane's losing her parents like that.   I'm still getting used to this possessive gerund structure. It sounded so weird to me at first. Is the structure used in both formal and informal contexts? Are there any alternative structures that result in the same meaning and are more frequently used? (Examples taken from )
Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$  According to the book [Dunford and Schwartz, Linear operators part I, page 453] $C$ is compact in the $\ell^2$-norm, but there is no proof. How can I show that $C$ is indeed compact in $\ell^2$ ? For a given ellipse, find the locus of all points P for which the two tangents are perpendicular.   I have a trigonometric proof that the locus is a circle, but I'd like a pure (synthetic) geometry proof. I am being flown out by a company for an on-site visit. Will I earn miles for this trip, or will the company? Which IP address do I use to access my server? I'm running a Samba server on my MacBook and I've also set up port forwarding to said server from my router.  My public IP is 106.51.x.x (according to google and a few other services) and my Mac's local IP is 192.168.x.x, but when I head to my router's configuration page, it says my external IP is 10.242.x.x.  This is what I don't understand, when connected to my home WiFi, using the 10.242.x.x address works just fine. But if I try to access the same over 4G, the connection times out.  As for the public address, neither WiFi nor 4G works.  Is there some setting on my Mac (I'm running MacOS Sierra) that I must alter to make it visible to the public? I just ran the following command in a directory with several files, some owned by root, some by a normal user. I got a strange message:  $ sudo chmod -rwx * chmod: example-file: new permissions are ----w----, not ---------   Why the heck does chmod refuse to clear all permissions, if a file had group write rights? What is the thought behind this behaviour?  And is there a way to force removing all permissions in one step, without having to execute sudo chmod g-w * afterwards? Is there a way to start a PC with Ubuntu from the Keyboard? Because my button &quot;Power&quot; is broken. Title page line spacing I have a simple problem but I didn't manage to find any answer for it. In my title page, I just used  \title{Name of the title}   then  \hrule height 0,1cm \leavevmode \\[0,5cm]     {\Huge{\textbf{\@title}}}\\[0,5cm] \hrule height 0,1cm \leavevmode \\[0,5cm]   to print the title of my document. But it takes more than 3 lines on the PDF file, and I have a different spacing between the 1st and 2nd line compared to the 2nd and 3rd, which is a bit frustrating.   Anyone got an idea of why ? Regain of reputation from downvoted and deleted answers - update FAQ Can I connect to a specific access point by BSSID? Is there any way to get Mac OS X to connect to a specific access point by BSSID? I have a network with the same SSID for multiple access points, and I would sometimes like to specify which specific access point to connect to. Is this something Mac OS X can do natively, or is there a third party tool that can do this? How can I record sound from the windows command line? Is there an application capable of recording sound that I can run from the command line on XP?  Ideally I'd start it with one command and stop it with another. Or I could specify a duration with the starting command. How do I get started working with JSON in Apex? Problem from Herstein on group theory Show that $13$ divides $2^{70} + 3^{70}$.   My main problem here is trying to figure out how to separate the two elements in the sum, and then use Fermat's little theorem. So how can I separate the two?  Thanks! Why aren't logarithms defined for negative $x$? Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$), are true, why aren't logarithms defined for negative numbers?  Why can't $b$ be negative? Take $(-2)^3 = -8$ for example. Turning that into a logarithm, we get $3 = \log_{(-2)}{(-8)}$ which is an invalid equation as $b$ and $x$ are not positive! Why is this? Short story about survivors on Earth after the atmosphere has frozen This is a short story I read maybe in the '80s.  Earth has been thrown away from the Sun, probably due to a very large object passing through the Solar System, and the atmosphere has frozen in layers. There is a small family - father, mother, two children, who live in a room they have made airtight by hanging many layers of material to slow the loss of air. They keep warm with a 'camp-fire'. The father goes out every day in a suit of some sort to find food and mine frozen oxygen. It ends when they are found by another, larger group of survivors.  Does anybody know the title and author? Root of unity for $n+1$ Let $n \ge 1$ and $u_{1},u_{2},...,u_{n}$ are all root of unity for $n+1$ and there are different from $1$.(a) Find polynomial $f \in \mathbb R[x]$ of degree $n$ for which: $f(u_{k})=0$ for $k=1,2,...,n$.(b) Calculate real coefficient and imaginary part for $(2-u_{1})(2-u_{2})\cdot...\cdot(2-u_{n}).$    I have a knowledge about root of unity but I completely don't know how to do it because I think that I need some extra trick to do it but my ideas finished. I know that $\varepsilon_n^{(k)} = \cos\left(\tfrac{2k\pi}{n}\right) + i\sin\left(\tfrac{2k\pi}{n}\right) = e^\frac{2\pi i k}{n}$ and for $n&gt;1$ $\sum _{{k=0}}^{{n-1}}e^{{\frac  {2\pi ik}{n}}}=0$ but I think it is too little knowledge to solve this task.Can you help me? Java: Prefix/postfix of increment/decrement operators? Windows vs Linux Local Time? Proof that if continuous function on $[0, 2]$ and $f(0) = f(2)$, then for some $x: f(x) = f(x - 1)$ Is it possible for an x86 processor to match an ARM processor in terms of performance per watt? From my personal experience with my tablet, and from the benchmarks and articles I've read, it always seems ARM processors, as seen in virtually all mobile devices, deliver incredible performance for the amount of power they consume.  It really seems like ARM is inherently more power-efficient than x86.  To me, it looks like that this is because x86 is a  while ARM is a . However, Intel Atom processors deliver very good performance relative to their power consumption as well.  In addition, having studied computer architecture, it is my understanding that in modern processors, the complex x86  is internally translated into simpler  that use an internal instruction set that is closer to the simpler ARM or MIPS ISAs than x86 proper.  As such, the ISA as presented to the programmer is little more than an interface to issue commands to the processor, rather than a representation of the actual low-level operations the chip performs.  Then again, implementing this translation layer requires additional silicon space on the chip...  That said, assuming that they are implemented using the same semiconductor process, is ARM inherently more efficient than x86? With a properly designed , is it possible for an x86 processor to deliver the same performance per watt as an ARM processor?  Remember, I'm looking for a well-written technical answer in the spirit of Stack Exchange and not mere speculation.