How can I prove this Euler method has a convergence order of two?
Numerical Approximation of Differential Equations with Midpoint Method
How are we able to calculate specific numbers in the Fibonacci Sequence? I was reading up on the Fibonacci Sequence,  $\text {{1,1,2,3,5,8,13,....}}$ when I've noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly simple, but I reckon I can't find any formula for calculating a Fibonacci number based on it's position.  Is there a way to do this? If so, how are we able to apply these formulas to arrays? View vote count button on mobile On the mobile site, can there be a button directly below the vote up/down buttons for viewing vote counts? When I am using the mobile site and I want to view vote counts, I have to switch to the full site to do it which is very inconvenient.   It could be a small link that says "vote counts," because I don't know what the button would be (the letter "V?" a "+/-" thing? I have no idea).  Related:  , but I am requesting an additional button below the vote buttons. That way you won't accidentally click a vote button. CHECK constraint in MySQL is not working I remember (vaguely) reading somewhere that it is wrong to use the present perfect when the subject of a sentence is someone who is long dead:     *Einstein has visited the Philippines.   My question is, What if the dead person is not the subject but an object:     Leo is the best president our country has/had ever had. (The ones who   succeeded him were inferior, even our current president.)   Which tense is proper here align vs equation Oxford Comma Conventions I'm using Rules 7.x-2.6 and the way it works has changed a bit since I last saw.  I have a rule as follows:  EVENT A node has been flagged, under "Report content"  CONDITIONS None  ACTIONS Send mail to [site:mail]   However, I'd like to, instead of sending directly to [site:mail], send an email to all users flagged with an "admin" role.   Now I know that I'll need to provide a loop do so, but my sticking point is generating the list of users to pass to the loop. So far, I have attempted to add a new action:   "Fetch entity by property" Entity type: User Property: User roles Value: admin   But this doesn't seem to give me what I want in data selection, when passing it to the loop. Can someone recommend what they would set here? I think the following is true:  If $q$ is coprime to $a$ then $a\mid (nq-1)$ for some $n\in\mathbb Z$.  I have started to first reducing $n\in\mathbb Z$ to $n\in\{1,\ldots,a-1\}$ and then considering all the cases $q = ka + j$ for $j=1,2,3,\ldots,a-1$. I stopped at $j=3$ and found that this is kinda messy. Does anyone have an elegant proof for this (maybe using some results from number theory)?  [Note: question has been generalized from prime $q\nmid a$ to $\,q$ coprime to $a$] Recently I'm working on a project aiming to convert the vibration of sound waves into electricity to charge batteries using a piezoelectric material (MFC type). I was trying to use the human voice as an input, but theoretically it was not enough, so do you have any ideas about some places that I can apply my project on it (that must be noisy or can give me vibrations)?  Moreover, has the MFC material been used before in this application? If I used many pieces of it in parallel would that help? Virtualbox: OS cannot use the new extended disk space (.vdi) How do I write an Apex unit test? Parsing XML, JSON, and newer data file formats in UNIX using command line utilities The Unix environment has some excellent tools for parsing text in various forms. However, of late, the data is not in the traditional (historical) formats (CSV, TSV, record-based or some other delimiter-based) it used to be before. Data these days is exchanged in structured formats like XML/JSON.  I know there are some good tools like sed, awk and Perl which can chew down nearly any form of data out there. However, to work with this sort of structured data, often one has to write a complete program, and, given the little time available to extract information, one has to sit down and figure out the whole logic of what one wants to query and put it down programmatically. Sometimes this is not OK – basically because the information extracted from those files acts as inputs for further work; also because of the time it takes to search for the appropriate solution and code it up. A command line tool is needed with sufficient switches to find, query and dump data.  I'm looking for tools that take a XML/JSON or other forms of structured data and dump it into other formats like csv, etc., so that from there one could use other commands to get any information out of it.  Are there any command line utilities you know of which do this kind of a job? Are there already awk/Perl scripts available to this? Proving $\sup B=\inf A$ Let $A$ be a nonempty and bounded below, and define $B= \{b\in \mathbb{R}: b$ is a lower bound for $A\}$. Show sup$B$ = inf$A$.   So far I have: Let $A$ be nonempty and bounded below. This implies $\exists l \in A, \forall a \in A, l\leq a$ and $l =$ inf$A$. Let $B = \{b\in \mathbb{R}: b$ is a lower bound for $A\}$ and let $M=$ sup$B$. This implies $\exists b\in B, \forall b \in B, b\leq M$. By definition of $B$, we know inf$A \in B$. Since $b\leq$ sup$B$, with sup$B$ the largest element in $B$, we have inf$A \leq M =$ sup$B$, so inf$A \leq$ sup$B$.  I realize I have to prove that sup$B \leq$ inf$A$ to get that sup$B$=inf$A$. I am having trouble doing so.   Part B: Use (a) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness. Holomorphic function has at most countably zeros Use of 'const' for function parameters I assume the difference in the number of Stack Exchange sites/communities is a mistake... correct? Making mathematical animations with Mathematica How can I show or hide boot messages when Ubuntu starts? I noticed that, for each research project, there comes a point where I have to consult the literature, and find all the previous research related to the current project.  Unfortunately, I seem to be very inefficient in my approach. I spend a lot of time digging through papers that are really interesting, but only of tangential importance to my current work, just to find the few papers that actually do similar things. (The pile of "cool paper, read when time" just keeps growing.)  Hence, I'm wondering: What is the most time efficient way to get a thorough overview of the literature in a specific niche of a field?    While searching for an answer I came across these questions:   - This question is quite similar and I liked the approach. I debated with myself if this is a duplicate, but I think it is focused more on how to be sure the search was exhaustive, whereas I'm worried about time efficiency. (Also, I'm a bit curious if things have changed since 2012.)   - The answer provides tips on how to identify what literature generally is important in a field, but I am looking for the most cost effective (least time consuming) way to do it in an area, where I know the broad literature, but not the specifics. This is a  about DNS server administration.   I have one hundred or so domains. All of these domains need to be configured identically, but it seems like a huge waste of time to have to configure a new zone and/or zonefile for every one of these domains. There has to be a better way to automate this!  I think I'm on to something...if I create a zone called ., or use some other feature in my DNS software to always return a specific IP when an A record is requested, this seems to get me pretty close to my desired end result. My server is responding authoritatively to the requests and it's so much easier to manage!  This was working great until nameserver validation software started checking these domains. I figured out that I can make most of the errors go away by adding NS records, but my software won't let me put more than one SOA record in the same zone file.  How do I work around this multiple SOA record problem?