Examples of functions $f:\mathbb Z \to \mathbb Z:\;$ surective, injective, bijective?
Surjective, Injective, Bijective Functions from $\mathbb Z$ to itself
Does the derivative of a continuous function goes to zero if the function converges to it? Physicist here. I am puzzled by a question: looking at a continuous function $g :\mathbb{R} \rightarrow \mathbb{R}$ that goes to zero at infinity, I am interested in the behavior of its derivative $\beta = g'$. Precisely, does it go to zero too? By writing on paper it looks like: $$ \beta(+ \infty)=\text{lim}_{x\rightarrow \infty}\text{lim}_{h\rightarrow 0} \frac{g(x+h)-g(x)}{h}$$ And if I can invert the two limits, I get what I expect: $\beta(+\infty)=0$, so I was curious about the hypothesis behind this permutation. Is the requirement of continuity enough? Thanks. How to use hook_node_presave to change field's value Basically, I want to put first name &amp; surname together into $node->field_par_contact_name  But it gives me warning. (See below)   function partnership_node_presave($node) {   if($node->type == 'partnership') {     $first_name = $node->field_par_contact_first_name['und'][0]['name'];     $surname = $node->field_par_contact_surname['und'][0]['name'];      // https://drupal.org/project/field_hidden     // Hidden variable     $node->field_par_contact_name['und'][0]['value'] = $first_name. ' '. $surname;      // Warning after save: array_flip(): Can only flip STRING and INTEGER values! in DrupalDefaultEntityController->load() (line 173 of .../drupal-7.23/includes/entity.inc   } }   Updated, please see image: Are C++ Templates just Macros in disguise? I've been programming in C++ for a few years, and I've used STL quite a bit and have created my own template classes a few times to see how it's done.  Now I'm trying to integrate templates deeper into my OO design, and a nagging thought keeps coming back to me: They're just a macros, really... You could implement (rather UGLY) auto_ptrs using #defines, if you really wanted to.  This way of thinking about templates helps me understand how my code will actually work, but I feel that I must be missing the point somehow. Macros are meant evil incarnate, yet "template metaprogramming" is all the rage.  So, what ARE the real distinctions? and how can templates avoid the dangers that #define leads you into, like   Inscrutable compiler errors in places where you don't expect them? Code bloat?  Difficulty in tracing code? Setting Debugger Breakpoints? Why has my database connection failed? I know it's possible to match a word and then reverse the matches using other tools (e.g. grep -v). However, is it possible to match lines that do not contain a specific word, e.g. hede, using a regular expression?   Input:  hoho hihi haha hede   Code:  grep "&lt;Regex for 'doesn't contain hede'&gt;" input   Desired output:  hoho hihi haha I'm reading .     The graph of $f: A \rightarrow B$ is the subset of $A × B$ consisting   of all ordered pairs $(a, b)$ with $f (a) = b$. If $A$ happens to be   $B^n$ for some $n ∈ N$, then we say that $f$ is a function on $B$ and   $n$ is the arity of $f$.   I don't understand the notation for the exponent of a set, I'm aware of what  is, it's the number of arguments or operands the function takes. I'm just not so sure if I can assume that a set with an exponent such as in $B^n$ means that a function has two operands.  I know the question may be too naive and perhapes the answer is the one I suggested and it's right in front of my face, I'm just searching for a confirmation or the negation of it. Thanks. Using getopts to process long and short command line options This is a generic inquiry on the common practice/standard for determining the acceptable values of RMS when georeferencing topographic maps.  Is there an absolute value?  Some literature suggests that it should be "less than or equal to 1/2 of the side of a cell which make up the total resolution of the image." How do I check if a string is a number (float)? I have a circuit board in production, primarily through-hole, to the tune of ~400 units per year. Each one is manually inspected and tested. The board has ten distinct voltage rails, four voltage feedback amps, three current feedback amps, a microprocessor with particular response characteristics, and a few other parts. That's a lot to check.  I'm considering building an automated test rig for all that. I think a PCB that feeds the appropriate input stimuli, observes the responses, and gives a go/no-go would save me a lot of time. I'm reasonably confident that this is done, commonly. However, the techniques used to do it are unknown to me.  I imagine a test rig with a circuit board of the same footprint as the unit under test (UUT). It would have a microprocessor, appropriate protection and scaling circuitry, and a large number of vertical pins sticking up from the board. Mount points would also stick up from the test rig for the UUT. When the UUT is mounted on top of the test rig, the bottom of the board makes contact with the vertical pins, allowing the test rig to feed stimuli to the UUT and observe responses.  This seems reasonable, and I have the idea I've seen something very much like this before. But the details elude me. What sorts of pins would be used on the bottom board? (Part numbers would be great!) Would I need to put specific kinds of test points on the UUT, or can I get away with just contacting the clipped pins protruding through the board? What other concerns might there be? Being characteristic is transitive Here, I wanted to verify that:     The property of being characteristic is a transitive relation among subgroups of a group $G$.   For subgroups $N,H\leq G$, we have $N$ char $G$ and $H$ char $N$. So $N$ char $G$ implies: $$\forall\psi\in Aut(G); \psi(N)=N$$ $H$ char $N$, so for all elements in $Aut(N)$, $H$ is remained never changing. Especially, when I take $\psi'=\psi|_{N}$ then $\psi':N\to Aut(N)$ would be in $Aut(N)$ and $\psi'(H)=H$. Since the last equality is true for $H$ and the maps, caused by restriction on $N$, then I have $H$ char $G$.  Honestly, I am inly not satisfied form the conclusion here and think I am losing something. Thanks for your hints. Normal ubuntu desktop will update ubuntu-desktop package and it's dependencies on release upgrade. Will Ubuntu minimal cd install ubuntu-desktop on release upgrade or just installed packages? I was recently in a bit of a quandary over which answer to accept to .  Of the answers I got, two were the clear leaders and, of course, I up-voted both. Choosing a 'winner' was more difficult though as their approaches were quite different.  Answer A addressed my question and provided a clear and well thought out response. I'd anticipate that this answer would be of most use to a user who found my question via a search. Selecting this would probably be the more community-focused way to go.  Answer B took a more lateral approach and made some guesses at the business logic behind my request, suggesting that I might be asking the wrong question. Their solution was based on this assumption - and they were spot on. Answer B formed the basis of my fix for the problem.  So... Best answer to the question or best solution to the problem - they may not always be the same thing.   I took the selfish approach and went with B. I'm not losing sleep over it, but I do wonder if I did the right thing? Showing that a sequence is monotone and thus convergent Here's the original question:     Let $(a_n)$ be bounded. Assume that $a_{n+1} \ge a_{n} - 2^{-n}$. Show   that $(a_n)$ is convergent.   Okay, I know that if I can show that if the sequence is monotone, I can conclude that it is convergent. But I am not sure how to show that it is monotone.  I know that  $$a_n \le a_{n+1} + \frac{1}{2^n} &lt; a_{n+1} + \frac{1}{n}$$  It looks to me as if it is monotonically increasing but I'm quite not sure how to prove my claim. Any hints would be appreciated. I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :) I've seen Breaking Bad complete series 2-3 times, but I never found out how Walter White was able to poison Brock.  I mean did he himself directly feed the Valley of the Lilly berries to Brock or did he let Saul do the work? I can't figure it out. Don't we have single sign-on for all Stack Exchange sites? I would like to use many sites in the Stack Exchange network, but I need to login to each site separately. Though it's easy, can't we solve this problem? I know each site has a different url :( but is there any way? Should individual letters be preceded with "an"? I was reading a comic book and I noticed that the author used "an A" and "an N" in a particular phrase (screenshot below). While I understand the usage of "An A" here (A being a vowel and what not) and "N" having a pronunciation which has a sound similar to "A".  Since most of English alphabets have pronunciations which start with a syllable similar to one of the vowels or semi-vowels; is it OK to prefix the letters using "An"?  For eg. An X, or An Y.        Zenescope's Grimm Fairy Tales - Werewolves: The Hunger, page #8 (Click to enlarge) How to share worlds between platforms with Minecraft Better Together I have read that with the new "Better Together Update" (BTU) it is now possible to .  However, I am having trouble finding details of how this is supposed to work to determine if it would do what I am hoping it would and therefore whether it is worth my time/money.  I have some saved worlds on Minecraft for XBox One, all done in creative mode.  I am able to import them into what is now just called "Minecraft" which as of the BTU is what is being used to mean the cross-platform version.   Can I now play this same world on my Windows 10 PC?   If I go to the Minecraft web site it offers to sell me the "Java Edition" with the addendum "includes edition for Windows 10".   But from what I have read, anything that has the word "edition" in it isn't the cross platform version.   It's very unclear to me what I would actually be buying here.  If it is possible to open the same world on different platforms, exactly how would I go about doing this?  On the XBox I have to sign in with an outlook.com e-mail address to access my worlds, or my kids' sign-ins to access their worlds, is it the same from the PC? And where would my world be saved?   I have read that XBox saves will be in the cloud, but PC saves will be local, so how can I get a cloud save onto my PC?  And then how can I get it back to the cloud when I am done so I can play it on the console again?  Will there be two copies floating around, making it possible for me to make a backup on my PC?  How will this interact with world editors?  At some point will I be able to start a game on XBox, transfer it to the PC, open it in a world editor, make some changes, and then transfer it back to the XBox? Let's say I'm running a simple server and have accept()ed a connection from a client.  What is the best way to tell when the client has disconnected? Normally, a client is supposed to send a close command, but what if it disconnects manually or loses network connection altogether? How can the server detect or handle this?