Applications of infinity in real life I am writing a mathematical essay and would like to focus on the concept of infinity. I am not sure of any real life applications of infinity to write about or some way to narrow down the topics. Does anyone have any ideas on ways to focus the essay on a section of the topic infinity.
Does the concept of infinity have any practical applications? I know what you're thinking: "of course it has, for example, it can be used to tell you how many times you can go around a circle". But that isn't really true, now is it? You'd be dead or the world would go under long before an infinite amount of loops had been reached.  Are there any practical applications for the concept of infinity? Is it a useful concept in maths at all?  I know that Donald E. Knuth has argued that for all practical purposes, a very, very large number has the same effect as infinity, in his book "Things a Computer Scientist Rarely Talks About" (can't remember the exact quote, nor find it online, unfortunately).  Examples are appreciated.
Pokemon Go Supported Devices How does Mage Armor interact with Unarmored Defense? Variable with colon dash ${VAR:-} What is the correct JSON content type? I've been messing around with  for some time, just pushing it out as text and it hasn't hurt anybody (that I know of), but I'd like to start doing things properly.  I have seen so many purported "standards" for the JSON content type:  application/json application/x-javascript text/javascript text/x-javascript text/x-json   But which one is correct, or best? I gather that there are security and browser support issues varying between them.  I know there's a similar question, , but I'd like a slightly more targeted answer. Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges. Add optional box for spam flags letting you say why you think it's spam , where a user had taken the time to look at the link on a post and realized that the post wasn't just off-topic, it was actually half-decently-disguised spam. So he flagged it as spam, and an overworked mod declined it, giving the questioner the benefit of the doubt. And we probably want mods to be giving the questioner the benefit of the doubt.  We also want to get rid of spam ASAP, and make the lives of mods easier. I'd like to suggest a feature the mod in question mused might be useful: An optional textbox where you can add a note saying why you think it's spam, so the mod can see it easily. In this case, the helpful user who went the extra mile could have written "Site linked doesn't relate at all to the question, instead solicits BitCoins" or similar.  One can leave a comment on the question, but:   People sometimes leave...unhelpful...comments on spam questions, and we don't want mods to have to take the time to read through them hoping for something useful It allows the user flagging the spam to do so and say why without calling attention to themselves publicly     Update: We may well want this for all flags, but in particular I had reason to want it for an NAA flag, so I've posted . I found a problem with an application on Ubuntu.   Questions :   How do I best report the issue?  What sort of information should I provide? Lock time not involved in MySQL slow query selection? I wish to analyze some read-write contentions problems on a very huge table where writes are very fast, less than 1ms in most case. The point is that the contention makes those write queries sometimes wait for the lock for a long time, i.e. hundreds of milliseconds. But the query itself remains very fast so it's not selected as a "slow query", which threshold is actually 10ms.  The only way to gather the information I need is to put à 0ms threshold so I can capture all queries, but it's not sustainable for more than 10mn on our production environment, where it would produce about 200MB of logs per minute.  In fact I can't understand why MySQL don't include the lock time, added to the execution time, for the slow query log selection process. At least, I wish there were an option for that.  Am I missing something ? How to make blocks in spawn unbreakable in survival? So on my multiplayer server on Minecraft Education Edition I can't seem to figure out how to make the blocks in spawn unbreakable. I know adventure mode fixes that but it's a survival world. I don't want them in adventure, so how do I make the blocks in spawn unbreakable? Is time continuous or discrete? I have the following sentence:     If T had still been alive, there is the great possibility that either T or C ...   My teacher says that the word "then" must appear after the comma, but I think that it's implied and unnecessary. Who is right? Hole-and-nail paradox in special relativity Automate fixes for issues found by Sonar $R^2$ of 1 but RMSE > 0 How do you call an external command (as if I'd typed it at the Unix shell or Windows command prompt) from within a Python script? Reference Text that develops Linear Algebra with Knowledge of Abstract Algebra Background: Due to some unfortunate sequencing, I have developed my abstract algebra skills before most of my linear algebra skills. I've worked through Topics in Algebra by Herstein and generally liked his approach to vector spaces and modules. Besides a very elementary course in linear algebra (where most of the time went towards matrix multiplication), I have not developed any other linear algebra skills.  But it seems that it is now necessary for me to do so. Most of the topics that I am looking at now require some background of linear algebra and I still lack understanding of ideas like: bilinear forms, invariant subspaces, eigenvalues, requirements for diagonalization of a matrix and so forth.  This brings me to my question (provided there are any)     Question: What are some good texts that develop the theory of linear algebra from a perspective of general algebra?   Are there any texts that develop the key (elementary) ideas of linear algebra in an abstract setting? Thank you for the help! Multiple file upload in php Multiplexing UART I'd like to interface my main microcontroller's hardware serial port to more than one device at the same time. My understanding is that UART on my microcontrollers is based on an active-low 5V TTL.  (I know what I'm explaining here can be technically achieved by using more suited protocols, such as SPI and I2C, but my project requires to be only implemented using UART.)  Here's a pseudo-schematics chart of what I'm trying to achieve:         &ndash; Schematic created using   Please note that:   I don't want to use a second UART interface on the master, or any of the slaves. Only one slave at a time is going to transmit or send data on the line to the master. (e.g. If the slave 1 is transmitting data, slave 2 and 3 should remain idle.) Whatever the master is transmitting can, and should be received by all of the slaves. All of the slaves should be hot-pluggable, meaning they could be removed or inserted at any time. The RX and TX wires to the master are parallel between each slave, so no daisy chaining.   So, my questions are as follows:   Assuming the correct impedance is considered, I shouldn't need an additional circuit to transmit data from the master's TX line to any of the slave's RX line simultaneously. Is this correct? I understand that without current limiting resistors between slaves' TX line and the master's RX line, if any of the devices begin to broadcast data, the line between the devices will short circuit and I risk damaging them. Is this correct? Assuming the items 1 and 2 are correct:  a) Would just some current limiting resistors be enough to use?  b) Can I use 74-series logic gates instead of resistors? Between options a and b above, what route should I choose? For a reliable 115200 baud rate connection, will the switching frequency of a 74HC08 AND gate suffice to be incorporated?   EDIT: I'm well aware of other well-suited protocols such as SPI and I2C. There's a reason I'm using UART, and that's because the optiboot bootloader that is used on the master's mcu communicates over UART, albeit UART's obvious limitations.  Also, while daisy-chaining is definitely a clever approach, due to breaking the 4th requirement, it also breaks bootloader functionality.  The devices could also be USB-to-Serial chips, and using SPI-to-USB or I2C-to-USB would not be feasible in this project.  So, I'd appreciate if you could focus on the 6 questions listed here instead, and share your UART knowledge. $\epsilon$ undefined using Minion Pro with pdftex I'm using MiKTeX 2.9 and have recently installed the Minion Pro fonts for use with pdftex. I followed the instructions available on  to install them.  Everything works as expected, but in place of the Greek letter epsilon I get a box with a cross (see below), and the following warning:    I found the issue was reported , but I still can't fix it. xelatex works fine, but I'd like to to stick to pdftex so I can use microtype.  \documentclass{article}  \usepackage[T1]{fontenc} \usepackage{MinionPro}  \begin{document}  Text here. $\epsilon$  \end{document}       UPDATE: a set of instructions for the installation of the up-to-date FontPro scripts with MiKTeX can be found . Prove that every number ending in a $3$ has a multiple which consists only of ones.