Is there a notation for addition form of factorial?  $$5! = 5\times4\times3\times2\times1$$  That's pretty obvious. But I'm wondering what I'd need to use to describe  $$5+4+3+2+1$$  like the factorial $5!$ way.  EDIT: I know about the formula. I want to know if there's a short notation.
(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems)  I'm aware of Sigma notation, but is there a function/name for e.g.  $$ 4 + 3 + 2 + 1 \longrightarrow 10 ,$$  similar to $$4! = 4 \cdot 3 \cdot 2 \cdot 1 ,$$ which uses multiplication?   Edit: I found what I was looking for, but is there a name for this type of summation?
Is there a term for switching syllables of words? Primary question: A common speaking mistake is to exchange syllables of words, saying "It's trace rhyme!" instead of saying "It's race time!", or pronouncing "kickin' chackatory" instead of "chicken cacciatore." Is there an existing name in poetry, linguistics, or English studies for switching syllables of (quasi-)consecutive words? I am not especially interested in the case when switching syllables results in interpretable homonyms or portmanteaus, like in the first example.  Secondary question: if not, what would you call it?  Tertiary question: is there somewhere a (playful?) taxonomy of fruitful or unfruitful lingual aberrations? GridBox/packed array behaviour Do sets, whose power sets have the same cardinality, have the same cardinality? Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$?  Why? Thanks. Am not doing anything fancy, just trying to update a field on Contact in JS of LWC but getting this exception.  Uncaught TypeError: 'set' on proxy: trap returned falsish for property 'Name' throws at mydomain/auraFW/javascript/mhontaYdOya4Y_lBu7v9yg/aura_prod.js:2:27687  HTML Code:   &lt;template&gt;      &lt;template if:true={wiredContact}&gt;          {wiredContact.Name}          &lt;lightning-input value={wiredContact.Name} onchange={updateName}&gt;&lt;/lightning-input&gt;     &lt;/template&gt;  &lt;/template&gt;   JS:   import { LightningElement ,wire,track,api } from 'lwc'; import myContact from "@salesforce/apex/ContactController.fetchContact";  export default class Myrefreshapextest extends LightningElement {    @track wiredContact;   @wire (myContact)         fetchedContact({error, data}){             if(data){                 console.log(JSON.stringify(data));                 this.wiredContact = data;             }else if (error){                 console.log(error);             }     }      updateName (event){         console.log(JSON.stringify(event.detail.value));         console.log(JSON.stringify(this.wiredContact));         this.wiredContact.Name = event.detail.value;     }  }   Apex:  public class ContactController {      @AuraEnabled(cacheable=true)     public static Contact fetchContact(){         return [SELECT Id,Name FROM COntact LIMIT  1];     } }   On top of my head, am not doing anything wrong, anyone has idea what's wrong with my code?  When I print console.log(JSON.stringify(this.wiredContact)); I get old values so am pretty sure it exists.  I tried with @track and @api, but same response. Can anyone shed some light? What should I look for when buying a used bicycle? I'm moving to a city where I can ride a bicycle to the grocery store, coffee shop etc. I want to purchase a decent used bicycle. I'm leaning towards a bike that's in between a "10-speed" or road-bike and a mountain bike. What should I look for when choosing a particular bike?  I've never purchased a used bicycle. Other than classifieds (e.g., Craigslist), what types of places can I look? What other issues should I consider, such as size, visual appearance, leaking tires, etc.? How can I make sure it's not stolen? List sections of chapter at beginning of that chapter How can one have a mini table of contents for the current chapter appear just after the chapter name listing all sections of that chapter? How can I view the contents of tar.gz file without extracting from the command-line? I want to see the contents (list of files and folders) of an archive, for example a  tar.gz file without extracting it.  Are there any methods for doing that? Is it unreasonable to assign a MySQL database to each user on my site? I'm creating a user-based website. For each user, I'll need a few MySQL tables to store different types of information (that is, userInfo, quotesSubmitted, and ratesSubmitted).  Is it a better idea to:  a) Create one database for the site (that is, "mySite") and then hundreds or thousands of tables inside this (that is, "userInfo_bob", "quotessubmitted_bob", "userInfo_shelly", and"quotesSubmitted_shelly")  or  b) Create hundreds or thousands of databases (that is, "Bob", "Shelly", etc.) and only a couple tables per database (that is, Inside of "Bob": userInfo, quotesSubmitted, ratesSubmitted, etc.)  Should I use one database, and many tables in that database, or many databases and few tables per database?    Edit:  The problem is that I need to keep track of who has rated what.  That means if a user has rated 300 quotes, I need to be able to know exactly which quotes the user has rated.  Maybe I should do this?  One table for quotes. One table to list users. One table to document ALL ratings that have been made (that is, Three columns: User, Quote, rating).  That seems reasonable. Is there any problem with that? What's the smallest positive integer which is divisible by all integers $1, 2, \dots, n$?  Is there a simple way to represent the answer?  Call it $f(n)$ here.   Clearly factorial ($n!$) satisfies the condition of divisibility.  But once you get to $f(4)$ (which is $12$), factorial ($24$) is too large, because it includes as factors both $4$ and $2$, which are redundant.  Multiplying all the prime numbers up to $n$ gives you another estimate, which is this time too low, because you need to include primes multiple times when they're repeated in the factorization of the inputs.  So the answer is somewhere between factorial, and the product of primes.  Is there a simple answer? Sponsor Gallery, Page, or Something Like That? I'd like to do some URL rewriting (Why? See .) so that instead of users seeing addresses like   labouseur.com/course-compilers.html    they can instead see and use simply   labouseur.com/course-compilers   (Even better, maybe I should restructure that so that it's courses/compilers.)  I'm using a Linux-based shared hosting service for my website, so I do not have administrative control of the server, but I do have control over .htaccess. The references I've read online seemed less than clear to me, so I'm looking for a little clarity and advice here. Examples of involutions on $\mathbb{R}$ Just recently I read up something about involutions (functions $f: A \rightarrow A$ such that $f(f(x))=x$, for all $x$ in the domain of $f$), and was wondering how many (if there is a small set of general functions) such involutions exist for $A = \mathbb{R}$, or maybe $A = \mathbb{R} - S$, where S is some set of points that would make the involution work if they were left out of $A$. In general I'm interested in real functions that are involutions, continuous or otherwise.  There were some examples I found , but any more would be certainly very interesting! LuaLaTeX, TeXLive 2016, standalone: undefined control sequence I am reading a paper and have questions about the details of the procedure described. I have read other papers by the same team but they don't explain too much about that procedure anyway. I think it might be common, but my supervisor doesn't know it too.  I am stuck and I want to get out of it. As a student working on my master thesis, can I email the contact author for the manuscript cold out, or should I ask someone to contact him for me? I would ask my supervisor but I don't want it to imply that I'm avoid taking initiatives when I could do it on my own. I'm not a physicist; I'm just curious about what would happen.  According to this video "" (at 2:50):     "Somebody decided to leave the detectors on but just not take any data. [...] So if it was also going to a magnetic tape - there was no magnetic tape loaded. [...] And what do you think happed? They got this. They got the diffraction pattern."   So the act of measuring (beeing concious of the data) destroys wave-form pattern?  If so lets consider these cases:   What if I'd throw a dice after the experiment to decide if I should look at the data or to destroy it? (e.g. look at the data if the number on dice is even) What if the information about measurement would need to travel longer period of time than the particle would need to hit the wall?    To above cases: Would it mean that information travelled instantly - faster than light or that "the particle can look into the future"? the (surely simplistic) model of a bipolar junction transistor one is taught in foundational physics course appears to be symmetric. - So, what is the difference between the collector and the emitter of a real BJT? If the transistor were symmetric one would not make this distinction...  Also:   Do BJTs have a 0.6V voltage drop like diodes? Are BJTs conductive in both directions, i.e. E-C and C-E?   Many thanks. Why is it dangerous to initialize weights with zeros? Is there any simple example that demonstrates it? I want to decrease the vertical space between the top of the page and the chapter heading. I have tried to follow the instructions for titlesec but I am not successful. I have tried the following two setups without any changes occuring.  First:  \documentclass[12pt]{report}   \usepackage[compact]{titlesec}   \begin{document}   \chapter{Characters}  The origin of the group determinant begins with Richard Dedekind in the late 1800's. He began to explore the idea after studying the discriminant in a normal field [A]. He made several observations about the group determinant, but was only able to prove some of them.   \end{document}   Second:  \documentclass[12pt]{report}   \usepackage{titlesec}   \titleformat{\chapter}[display]   {\normalfont\huge\bfseries}{\chaptertitlename\ \thechapter}{20pt}{\Huge}   \titlespacing{\chapter}{0pt}{0pt}{0pt}   \begin{document}   \chapter{Characters}    The origin of the group determinant begins with Richard Dedekind in the late 1800's. He began to explore the idea after studying the discriminant in a normal field [A]. He made several observations about the group determinant, but was only able to prove some of them.   \end{document} How to connect Selenium to an existing browser that was opened manually in java or create semi automated tests? I want to achieve some use cases in which:  I want to perform some tasks that require manual intervention before kicking in automation execution. For instance, I need to manually launch the browser, navigate to the desired page, perform some manual tasks like entering captcha manually and then run the automation script to continue from there.  Debugging – Let's say we are working on a test script for a certain page (product checkout) in the application. But the script to reach that page (add to cart) is not ready yet. We can manually launch the browser and navigate to that page and then run the automation script.  Can someone help me to achieve this type of scenario? How is prototyping done in a professional EE environment? Do you do your prototypes on breadboards, on copper clad boards, Manhattan style, or do you just draw schematics, order PCBs made and assembled and work with them, hacking here and there if necessary?