How to create interface with overloaded arithhmetic operators in C#?
Arithmetic operator overloading for a generic class in C#
Does the rank-nullity theorem hold for infinite dimensional $V$? The rank nullity theorem states that for vector spaces $V$ and $W$ with $V$ finite dimensional, and $T: V \to W$ a linear map,  $$\dim V = \dim \ker T + \dim \operatorname{im} T.$$  Does this hold for infinite dimensional $V$?  According to , the statement is false.  But according to , page 4, the statement is still true.  I'm thoroughly confused. As I understand, correct plural versions of passerby and attorney general become passersby and attorneys general.  But why?  With passerby, the the preposition "by" has been combined with the noun "passer" to become a one word noun "passerby".  So why shouldn't this be passerbys?  If I am an attorney general, and you are an attorney general, why are we not attorney generals?  As an American I have always heard it as attorneys general, however  suggests British English does use "attorney-generals" (as well as an interesting 2nd paragraph that further probes my question).  Are there hard rules to this topic?  Or merely generally accepted practices based on locale? Convergence in Distribution in Order Statistics Let $X_1, X_2, \ldots$ be iid from Exp$(\theta)$ with density function $f(x) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$. (a) Find the limiting distribution of $M_n = Y_1 - \theta\ln(n)$ and $T_n = nY_n$, where $Y_1$ and $Y_n$ represent the first and last order statistics (among $n$ observations) respectively. Answer: I have obtained the CDF of X to be: $$F_x(x) = 1 - e^{-\frac{x}{\theta}}$$ the CDF of $Y_1$ to be: $$F_{Y_1} = 1 = e^{-\frac{xn}{\theta}}$$ the CDF of $M_n$ to be: $$F_{M_n}(x) = \left[1 - \frac{1}{n}e^{-\frac{x}{\theta}}\right]^n$$ with limiting distribution $$e^{-e^{-\frac{x}{\theta}}}$$ the CDF of $T_n$ to be: $$F_{T_n}(x) = 1 - e^{-\frac{x}{\theta}}$$ which is also the limiting distribution. (b) Show that $M_n$ and $T_n$ are asymptotically independent. Answer: I've tried finding the joint PDF for ordered statistics by using the equation $$g_{ij}(y_i, y_j) = \frac{n!}{(i - 1)!(j - i - 1)!(n - j)!}[F(y_i)]^{i - 1}[F(y_j) - F(y_i)]^{j - i - 1}[1 - F(y_j)]^{n - j}f(y_i)f(y_j)$$ however, it is a mess based on my CDFs. Any suggestions on a different approach? Proof Verification: Converse of Intermediate Value Theorem A function $f$ is increasing on $A$ if $f(x)\leq f(y)$ for all $x&lt;y$ in $A$. Show that the Intermediate Value Theorem does have a converse if we assume $f$ is increasing on $[a,b]$.  The converse of IVT along with the additional hypothesis would look like this: Lef $f:[a,b] \rightarrow \mathbb{R}$ be an increasing function on $[a,b]$ which satisfies: if $L$ is a real number such that $f(a)&lt;L&lt;f(b)$ (or $f(a)&gt;L&gt;f(b)$), then there exists a point $c\in (a,b)$ where $f(c)=L.$ If the preceding conditions are met, then $f$ is continuous on $[a,b]$.  The proof (if it's correct) I'm going to show is only for the $c \in (a,b)$, but I think for end points ($a$ or $b$), the method should be similar.  We want to show that given $c \in (a,b)$, for all $\epsilon&gt;0$, there exists a $\delta&gt;0$ such that $|x-c|&lt;\delta \implies |f(x)-f(c)|&lt;\epsilon$.  Since $f$ is increasing, we know that $f(a)\leq f(c)$. If $f(c)-\epsilon&lt;f(a)$, then set $x_1=a$, if $f(c)-\epsilon\geq f(a)$, then we know by hypothesis that a $x_1&lt;c$ exists such that $f(x_1)=f(c)-\epsilon.$ In either case, we have for $x \in (x_1,c]$ $$ f(c)-\epsilon \leq f(x_1) \leq f(x) \leq f(c) .$$  In a similar way, since we know that $f(c)\leq f(b)$, we can deduce that a $x_2&gt;c$ exists such that, for $x \in [c,x_2)$ $$f(c)\leq f(x)\leq f(x_2)\leq f(c)+\epsilon .$$  Pick $\delta = \min \{c-x_1,x_2-c \}$, we then have $$|x-c|&lt;\delta \implies |f(x)-f(c)|&lt;\epsilon .$$  Is this correct? In the proof that I saw, they originally used $\frac{\epsilon}{2}$, instead of just $\epsilon$, but I think this one works just as well, what do you guys think? How do I join two lists in Java? How can I lock a folder so that it CANNOT be erased? On my computer running Windows XP Pro SP3 I have a folder on my desktop that is very important. How can I lock it so that I can alter anything in it, but the folder itself cannot be removed without resetting a (lock/unlock?) switch? Can multiple render systems be used? I am designing a hobby project that will run on a battery so I am trying to reduce power consumption. The SoC I am using has the ability to go into a deep sleep mode where it consumes very little power, and then can be interrupted by a falling edge on a pin.  I want to generate a high-low-high pulse when the state of a reed switch changes in either direction. I have been reading a lot and it seems that a one-shot monostable circuit can be built from a couple of transistors and an RC circuit to generate an output pulse when an edge is detected. I understand how that works. However, I can't figure out the best way to also pulse on the opposite edge.  One thought, was to just combine two one-shots that are configured to trigger on rising and falling edges respectively, and use the transistors at the output to pull the common pull-up low (kind of like a NOR gate).  Since I am a hobbyist, I am mostly using passive components. What is the "real" way of doing this in a low-power configuration using logic devices?  Edit: I figure I should mention something about the timing requirements. Normally the reed switch will be either open or closed for many seconds at minimum before changing states.  Edit: Here is an example of the circuit I was using for detecting only the falling edge:    And here is the ~20ms pulse it was producing for ~200ms button press simulating the reed switch (Channel 1 is Vbe for T1 and Channel 2 is measured at RST): Command not found $ (dollar sign) Any command that starts with $ returns  command not found   How can I fix this?  I am  using 16.10 Ubuntu. What should I do if a question has changed since I answered it? The other members vote my answer down because it actually doesn't answer the current question.  This kind of editing sometimes happens with  where the OP keeps adding new questions, making my pre-existing answer now incomplete.  But another pattern is when the OP asks an "XY problem" question, an answer points it out, and he changes the question to react to that.  Those answers now not only aren't answers but can look very tangential to people who don't review the edit history. Preposition before noun phrases I have asked the same question in ELL stackexchange, but unfortunately haven't received enough answer/comment. The one answer I got is not satisfactory. So that's the reason I am asking it here again. .  I have seen in some cases prepositions are omitted before some noun phrases. And it's explained that those phrases are actually an adverb phrases. But I know a simple thing. If the head of the phrase is a noun, it's a noun phrase. If the head of the phrase is an adverb it is an adverb phrase. For example -   Look both ways before crossing the road. [both ways is a noun phrase, where the head is ways, but still there is no preposition.]  He approached me in a friendly way. [a friendly way is a noun phrase, where the head is way, but as expected unlike sentence #1 it's preceded by the preposition in. And I have never seen this phrase is used without a preposition. I believe dropping the preposition is wrong, according to the grammar.]  She made a pickle a different way from her mother. [a different way is a noun phrase, where the head is way, but strangely there is no preposition before it. But I have seen examples of a different way used both with prepositions and without prepositions. I think the preposition here is optional.]    Now from these example sentences I have tried to demonstrate my problem/confusing area. My question is - 1. When a noun phrase is used as an adverb phrase? 2. When before a noun phrase the placement of preposition is obligatory (like sentence #2)? And where it's optional (like sentence #3)? And where placing the preposition is wrong (like sentence #1)? How to install and run CPUSim 4.0.11 on Ubuntu 18.04 LTS? I'm going according to the steps mentioned on the cs colby .  First, I installed java on my system by typing the following commands-  sudo add-apt-repository ppa:linuxuprising/java sudo apt-get update sudo apt-get install oracle-java10-set-default sudo apt install oracle-java10-set-default     Then, according to the instructions, I downloaded the .zip file, extracted it and then in the terminal typed   java -cp .:richtextfx-fat-0.6.10.jar:reactfx-2.0-MR.jar -jar CPUSim-4.0.11.jar   But this didn't work.  of command 1 terminal.  Here is a screenshot of all the files that are there in the CPUSim folder,   On observing the names of the folders present, I thought that the previous command was wrong. So I tried the following ones,  java -cp .:richtextfx-fat-0.6.10.jar:reactfx-2.0-M4.jar -jar CPUSim-4.0.11.jar java -cp .:richtextfx-0.6.10.jar:reactfx-2.0-M4.jar -jar CPUSim-4.0.11.jar   But none of them seems to work. Now when I click on the CPUSim-4.0.11.jar file, nothing happens. I have just switched to Ubuntu from Windows 10, and have a very little knowledge about how things work here. Please tell me how I could install and run CPUSim 4.0.11 on Ubuntu 18.04 LTS. iOS navigation bar item image size Avoiding 'instanceof' in Java When solving this issue (  ), I did a function so that the plugin would auto-hide from all users except one (me:).  The flaw is that it only auto-hides when active.  At first, it was a mu-plugin, but I had to enable/disable it manually. It really doesn't need to run all the time, as it may slow the site (?) or even contain some sensitive manipulation of the site's language (another question mark).  But mainly the motivation is that I don't want the client activating/deactivating/deleting this plugin (is for my own use), and don't want it cluttering her own plugins list.  So, decided to split the functions in two plugins:   One is a Must-Use used to hide the regular one.  The other will be activated/deactivated upon necessity and is the one from the above linked issue.   But now I run into another question: how to hide one plugin from the Must-Use listing? how to get the string-lengths stored in array (C) Let's say we have a Python script my_script.py that does some data processing with Blender. This script accepts some arguments (e.g. arg1 arg2). How can I forward arguments to the script when using it with the CLI?  blender --background --python my_script.py   The only trick I could find was :  echo "arg1 arg2" &gt; my_input.txt &amp;&amp; blender --background --python my_script.py   Is this really the only way to do that?  Update  I found a  on Stack Overflow. I have a log file being written by another process which I want to watch for changes. Each time a change occurs I'd like to read the new data in to do some processing on it.  What's the best way to do this? I was hoping there'd be some sort of hook from the PyWin32 library. I've found the win32file.FindNextChangeNotification function but have no idea how to ask it to watch a specific file.  If anyone's done anything like this I'd be really grateful to hear how...  [Edit] I should have mentioned that I was after a solution that doesn't require polling.  [Edit] Curses! It seems this doesn't work over a mapped network drive. I'm guessing windows doesn't 'hear' any updates to the file the way it does on a local disk. Why do I always hear remote train horn at night? This is definitely not an illusion, as many people have the same experience. I have usually lived in places miles away from train stations, which makes it unlikely to hear any train horns during the day. However at night, occasionally train horns can be heard. I hope if any one can explain the physics of this effect, like possibly sound travels faster at low temperature. How can I calculate how long it will take to transfer a file over LAN? I want to know about the method with which we can calculate data transfer speed.   If I am sending a 1 GB file through a 1 Gb/s LAN connection, from one computer to another, how much time will it take to transfer that file?