Correctness of "of my sending the picture"
When is a gerund supposed to be preceded by a possessive adjective/determiner?
Let $A$ and $B$ be $n\times n$ matrices such that $ABAB=0_{n\times n}$.  Can we conclude that  $BABA$ must be  $0_{n\times n}$?   I fail to give a counterexample when $n=2$， so I guess the answer is "yes" Variable amount recurring billing Hi i'm looking for a payment gateway that can do recurring billing that changes month to month.  Fogbugz do this, they charge based on how many active users there were that month. All of the APIs that i've found only let you set a fixed amount and it's difficult/impossible to vary the subscription amount month to month.  Has anyone come across any services that do this? Radiation– white vs black house, hot or cool? Can a black hole be explained by Newtonian gravity? In the simple explanation that a black hole appears when a big star collapses under missing internal pressure and huge gravity, I can't see  any need to invoke relativity. Is this correct? "Didn't use to get" or "Didn't used to get"? I just made a minor , and  the grammar correction was unnecessary.  The original text was     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't used to get.   It was corrected to:     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't use to get.    Was this grammatical correction an improvement? Was the original text grammatically correct? Is the edited text grammatically correct? Find n such that $n^5 - n$ is divisible by $120$. (*Here $n$ belongs to the natural number set)  Seems a trivial problem, but the answer mentioned in the text book seemed incomplete to me so I'm here just to ensure whether approach to this problem is correct or not.  So my approach is as follows: We know that $n^5 - n$ can be written as $n^5-n = n(n-1)(n+1)(n^2 + 1)$  Now since it's product of three consecutive natural numbers so it's divisible by $3! = 6$. Now using Fermat's Little theorem we can prove that $n^5= n \mod5$ hence $n^5 - n$ is divisible by $5$ as well. So $n^5 - n$ is divisible by $30$. Now to ensure their is another factor of $4$ in the term we need to ensure that $n^5 - n$ is divisible by $8$. I found that all odd natural numbers and multiples of $8$ makes $n^5 - n$ divisible by $8$, hence $n^5 - n$ is divisible by $120$. But in the text book they have only mentioned odd natural numbers as answer, which seemed incomplete to me.  Thanking everyone in advance:P How do we control web page caching, across all browsers? There are a few examples of pairs of words ending with -ee/-er like employee and employer or advisee and adviser. What I was curious about is if there was any rule that would describe the relationship of the objects in a pair like this and situations when it's appropriate to create a counterpart for a given word.   I'll give you an example. It's relatively common in the computer programming world to see the word dragee, which describes an object that is being dragged with a mouse. I understand that this is a relatively new word and could not be found in any dictionary (I've tried). Is that acceptable to make up words like this one or is it just bad English? I am trying to proof the theorem that states that:  "A map to a smaller dimensional space is not injective"  So I first suppose that V and W are finite-dimensional vector spaces such that dimV > dimW and I let T be the linear transformation from V to W.  Given the above, you have that:  dim(null(T)) = dim(V) - dim(range(T)) >= dim(V) - dim(W) > 0  So I understand the first and second equalities. However, I don't understand why dim(V) - dim(W) is strictly greater than 0. I want to executed command from file (which is script) in current shell in terminal.  Example: I have file ch_dir contains following:  #!/bin/bash cd /usr   Now if I execute file usually as following then it executes commands in different shell:  pandya@pandya-desktop:~$ ./ch_dir pandya@pandya-desktop:~$   Here cd /user is executed in different shell.  But I want to execute in Current shell. How do I do that?  Note: Here cd is only used to explain. Question's aim is to execute any command in current shell How to find the mime type of a file in python? What does size of a directory mean in output of 'ls -l' command? Given a Lie group, how are you meant to find its Lie algebra?  The Lie algebra of a Lie group is the set of all the left invariant vector fields, but how would you determine them?  My group is the set of all affine maps $x \rightarrow A.x+v$ from $R^n$ to $R^n$ under function composition. Is there a term for a foreign word that looks like an English word but has a completely different meaning? examples:         Gift (German) = poison   poisson (French) = fish   embarazada (Spanish) = pregnant   sauce (Spanish) = willow   triviale (Italian) = vulgar   parentes (Portuguese) = relatives   slim (Dutch) = smart What theorems can I use to prove this? In a bash shell script, writing a for loop that iterates over string values How to move a file? How can I show a supremum of a set is also its limit point? It just seems like I can't wrap my head around this. For a set $A$ how can I show that the supremum of that set $\sup(A)$ is also a limit point of said set? Can I just set $$a=\sup(A)$$ $$b\le a \: \:  \forall \: b \in A$$ and then show that there's a point different from $a$ in any $\epsilon$-neighbourhood for any $\epsilon&gt;0$? Or should I use a completely different approach? I (think) that I've configured a static IP address in /etc/network/interfaces:  # The primary network interface  auto eth0  iface eth0 inet static     address 10.1.1.2     netmask 255.255.255.0     network 10.1.1.0     broadcast 10.1.1.255     gateway 10.1.1.1   And I change the file /etc/resolv.conf to include the desired nameservers.   Then I restart networking sudo /etc/init.d/networking restart  But after some hours resolv.conf always reverts back to the DHCP nameserver.   I should also say that this is Ubuntu Server with only l0 and eth0 interfaces. Prove that $f\in L^1(A)\Leftrightarrow \sum_{n}^{\infty}m(\{ x\in A : f(x)\geq n \}) < \infty$