Let $f: \mathbb{R} \to \mathbb{R}$ be continuous and 1-to-1. Show that for each borel $E$, $f(E)$ is borel.   Since $f$ is continuous it is measurable. So if $M $ is borel, $f^{-1}(M)$ is borel too. However, we are not working with pre-images here. It is clear that $f (f^{-1} M) = M$ due to $f$ being 1-to-1 but I'm sure we can not characterize all borel sets of the domain by $f^{-1}(M)$.  I'm stuck here. Any help will be appreiated
Suppose $f: \mathbb{R} \to \mathbb{R}$ is a continuous function which is 1-1, then does $f$ map Borel sets onto Borel sets?
Run faster than outer turret attacks As the title says, is there any way to run faster than the turret attack (the missile like thing) so that you can arrive spawn before the attack reaches you, and therefore can't kill you? You know, when you have low health, and you are targeted by the turret? Is there any item sets or things like that? If $q$ is coprime to $a$ then $a\mid nq-1,\,$ so $q$ is invertible mod $a$ What's the difference between convolution and crosscorrelation? What's the difference between convolution and crosscorrelation? So why do you use '-' for convolution and '+' for crosscorrelation? Why do we need the "time reversal on one of the inputs" when doing convolution? how do i change the color of typeface on photoshop? If we have a point charge and outside of it we have a non-conducting Gaussian sphere, then Gauss's law says that the net flux should be zero. I agree that the total field lines coming in are equal to the number of field lines coming out, but isn't it true that the magnitude of the electric field at those segments of the sphere where the field is coming in is greater than those where it is coming out? So when we integrate, wouldn't these areas, having a greater electric field, contribute more to the net flux? I don't see why they would cancel each other out. Norm and unit sphere I want to dynamically create a sequence of strings by manipulate an array of elements and create some arithmetic procedure.  for name in FIRST SECOND THIRD FOURTH FIFTH; do     $name = $(( $6 + 1 ))     $name = "${$name}q;d"     echo "${$name}"; printf "\n" done   The desire outcome would be the below for $6 equals 0.  1q;d 2q;d 3q;d 4q;d 5q;d   But I get this error  reel_first_part.sh: line 18: FIRST: command not found reel_first_part.sh: line 19: ${$name}q;d: bad substitution reel_first_part.sh: line 18: FIRST: command not found reel_first_part.sh: line 19: ${$name}q;d: bad substitution reel_first_part.sh: line 18: FIRST: command not found reel_first_part.sh: line 19: ${$name}q;d: bad substitution   I guess it's something simple. It used to work when I did something like   FIRST=$(( $6 + 1 )) FIRST="${FIRST}q;d" The classic example... $X \sim Po\left (\lambda\right ), Y \sim Po\left (\mu\right)$, X and Y are independent.  Show that the conditional distribution of X is binomially distributed. Or in other words, $P(X=k\mid X+Y = n) = P (\tilde{X} = k), \tilde{X} \sim B\left (n ,\frac{\lambda}{\lambda + \mu}\right )$.   I've so far managed to reach to this step, and have been stuck since. Just somehow gotta get a $\frac{1}{n!}$ in the denominator, that would then complete the proof..or at least I think..  $$P(X=k\mid X+Y=n) = \frac{\frac{\lambda^{k}\mu^{n-k}}{k!(n-k)!}}{P(X+Y = n)}=  \frac{\frac{\lambda^{k}\mu^{n-k}}{k!(n-k)!}}{\sum_{i=1}^{n} \frac{\lambda^{i}\mu^{n-i}}{i!(n-i)!}}$$  Thanks for the help! Substitute for onions and garlic I love the taste of onions and garlic and it seems lots of other people do too.  But they upset my stomach so much that I can't really cook with them.  What can I use in their place to give my food a similar flavour? I know it is possible for a group $G$ to have normal subgroups $H, K$, such that $H\cong K$ but $G/H\not\cong G/K$, but I couldn't think of any examples with $G$ finite. What is an illustrative example? What does it mean to "program to an interface"? Computational complexity of Fibonacci Sequence Many To Many Relationships I am working on a conceptual model for a database. I have the following entities:  Student (id, name, date of birth, department)  Subject (id, name)   This is a many-to-many relationship, one student may be enrolled in many courses, each course has many students. From what I understand, as I should break this many-to-many relationship by introducing a new into a entity, say 'registered' which allows a one-to-one relationship or one-to-many relationship?  Is this correct design decision? Uncaught TypeError: 'set' on proxy: trap returned falsish for property Name 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? Why are my quotation marks causing an “unexpected token” error with this command? The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$ When you search on any of the SE network sites, with a tag (so that a description is shown); it shows the tag info content box, with the highlight-search-term span elements.  For eg. I searched ; ; and the results are as shown:                  Notice the &lt;span class='search-highlight'&gt; boxes around searched tag terms. Problem: We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of elements in each row is zero. Prove that $\det(AA^T)=nk^2$, where $k\in \mathbb Z$.  What have I considered so far:   First I thought, since sum of all elements in each row is zero, zero is eigenvalue of $A$ but $A\in M_{(n-1)\times n}(\mathbb Z)$ confused me. I see that $AA^T$ will be $(n-1)$ by $(n-1)$, so I tried calculating $AA^T$ but I failed to see any connection.   I did some research and found  question but we had no mention of it, which makes me believe I am on the wrong track.  Thank you all for your help. In Python, how do I determine if an object is iterable? How do I restrict scaling to a specific axis? Im new to Blender but have some experience with 3dsmax. In blender, when i select some polys and try to scale them, i have no control on the amount and especially relative axis on which the scale would happen, where as with 3ds max i could scale restricted to x, y, z or any combination of using the handy widget. Is there a way to replicate this, or to get any control over the scale ?  thanks,