I need help solving this task, if anyone had a similar problem, it would help me. The task is: Determine all solutions of the equation in the set of complex numbers. $(1-z)^5=z^5$ I thought I could include it $z=x+iy$, but I see that it leads nowhere. Thanks in advance !
The question says to solve this equation: $(z+1)^5 = z^5$  I did. Just want to find out if I did it properly and if my run-around logic makes sense.  First I begin my writing the equations as:  $$ (z+1)^5 = z^5$$ $$ \mathbf{e}^{5 \mathbf{Log}(z+1)} = \mathbf{e}^{5 \mathbf{Log}(z)} $$ So $$  \mathbf{Log}(z+1) = \ln|z+1| + \mathbf{Arg}(z+1)i $$   $$  \mathbf{Log}(z) = \ln|z| + \mathbf{Arg}(z)i $$   Now, because the natural logarithm is one-to-one, I write:  $$ \ln |z| = \ln |z+1| \Rightarrow |z| = |z+1|$$   So assign $ z = a +bi$   So that $|z| = \sqrt{a^2 +b^2} = |z+1| = \sqrt{(a+1)^2 +b^2} \Rightarrow a^2 +b^2 = (a+1)^2 +b^2 \Rightarrow a^2 = (a+1)^2 \Rightarrow a = -\frac12 $  So, $z = -\frac12 + bi$ and $z+1 = \frac12 + bi$ for some $b \in \mathbb R$  Now to find $b$  $$ \mathbf{Arg}(z+1) = \mathbf{Arg}(z)$$ $$ \tan^{-1} \frac{b}{\frac12} = \pi - \tan^{-1}\frac{b}{-\frac12}$$  I have a feeling this last part isn't quite right, so I just want to find out if I'm approaching this question properly?  Ultimately, I get $ z = -\frac12$ which upon inspection...is wrong...
How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$? Had it not been vs If it were not for How can I fit my table of contents into a single page? When I use \tableofcontents I get all my contents in a single page except for a line that goes on the other page, how can I make all in the same page? The statement is:     In a commutative ring with 1, every proper ideal is contained in a maximal ideal.   and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset  A\mid A\text{ is an ideal}\} $, then by set inclusion, every totally ordered subset has a bound, then $P$ has a maximal element $M$.  My question is why $M$ must contain $I$? Make display of reputation in revisions list consistent with the user card display The user card in the revision list, for example , shows the exact amount of reputation:  This is not consistent with how the flair works on (all?) other places in the site, e.g. under a post. There, the amount is displayed in units of thousands (k):  (or, in , millions (m).) It used to be consistent, see e.g. : How to upload a file of size 2GB from Apex? Is there any sobject that I can make use of? Expression Functions for Lightning Web Components How can expression functions similar to Visualforce and Lightning Aura Components be achieved within lightning web components?   For example, &lt;template if:true={index % 5 == 0}&gt;&lt;br&gt;&lt;/template&gt; expression index % 5 == 0 is not compyling at all.  accountList.html  &lt;template&gt;     &lt;template if:true={accounts.data}&gt;         &lt;template for:each={accounts.data} for:item='item' for:index='index'&gt;             &lt;!--LWC DOES NOT SUPPORT EXPRESSION FUNCTIONS--&gt;             &lt;!--&lt;template if:true={index % 5 == 0}&gt;&lt;br&gt;&lt;/template&gt;--&gt;             {item}         &lt;/template&gt;     &lt;/template&gt; &lt;/template&gt;   accountList.js  import {LightningElement, wire} from 'lwc'; import getAccounts from '@salesforce/apex/TestController.getAccounts';  export default class AccountList extends LightningElement {      @wire(getAccounts) accounts;  }   Since LWC doesn't support Expression Functions, I'd like to know how to implement them. Data Dump of Attachments I am trying to do an export of approx 30,000 attachments in our Salesforce Org,  Would anyone have been through a similar exercise and can advise best practice or the best way / tool to do this?  Kind Regards  matt I know there have been a lot of posts on this topic, but I haven't found exactly what I am looking for.   I noticed some dust spots on my sensor for my D750 after I took a picture with some sky in it and then was playing around with "dehaze" in Lightroom. Okay no problem, I have owned the camera for a year probably time to clean the sensor.   So I do a bit of research online and land on the fact that I can do this myself. I also want to learn how so I don't need to spend $50 or $75 each time I need this done. I decide to buy .  What I do is close my aperture down to F22, put my ISO on 100 and then set my shutter speed to 2 seconds or so and take a picture of a well lit white piece of paper.   I then go into Lightroom and jack the dehaze and look for dust spots. I see a ton. Okay, so now let's clean it. I wet the swab with a few drops to get it just saturated on the top of the swab, apply some pressure, wipe left to right, turn it over, wipe right to left and done.   Then I took the same picture and although the dust spots are different, they are still there! What the heck? Okay maybe it just needs a good blow out. I took my  and blow out everything I can inside of there, then I do the same cleaning method as the first time, except this time I start with a dry swab, throw it out, then get a new swab, wet it and clean it again.  I take the picture again... and they are STILL THERE. Okay let's get out the high power flashlight. I wouldn't say I scrubbed it, but I went back and forth with the swab on what seemed to be a stubborn spot (I know, sketchy). Then I clean it again with a normal swab and take another picture.   This went on and on and I have basically ran out of swabs. I still see spots. I used the same swab a few times (I know, also sketchy) because I just couldn't figure out why it wasn't getting clean. I even tried a can of air (although some don't recommend that because they have solvent in them, but I was desperate and some people on youtube did it). That actually seemed to help, but in the end I still have spots.  So obviously a lot of you will tell me to just "take it in! you're going to break your sensor", but I really want to learn how to do this. And now I am just stubborn.  The spots are at the top and bottom of the frame mostly.  Any ideas? I ordered  today, and I was going to try it out very carefully. I also ordered .       EDIT: I tried the gel stick. It did seem to help, but on the second try something else got on the sensor so I am going to wait until tomorrow to get the swabs and then try to do a swab cleaning + the gel stick again... we will see.  EDIT2: Well it looks . I think the gel sticky stick helped but it's still not spotless. I ran over it one more time. I also included a photo above to the people are nervous about cleaning their own sensor. Obviously I can't guarantee you won't screw up, but I ran a swab over this sensor at least 15 times and it still shots great. It is common for public Wi-Fi access points to include login pages, even at times, when they’re free (i.e. Starbucks, many airports, hotels, etc).  I’m often in a situation where I’ll open up Chrome/Firefox, with and all my previously open tabs will start to reload.  If there is a wireless login redirection page, however, every tab is redirected and reloaded and the previous page is lost from history (and the back button won't return to it).  Is there any way to avoid or circumvent this behavior? "Mutable variable is accessible from closure" in a function passed to Array.prototype.every lib32bz2-1.0 missing on 16.04 I have tried installing this package (needed for android studio).  This is the error i'm getting  E: Unable to locate package lib32bz2-1.0 E: Couldn't find any package by glob 'lib32bz2-1.0' E: Couldn't find any package by regex 'lib32bz2-1.0'   I have already tried:  sudo dpkg --add-architecture i386 sudo apt-get update sudo apt-get install lib32bz2-1.0   &amp;  sudo dpkg --add-architecture i386 sudo apt-get update sudo apt-get install lib32bz2-1.0:i386   I also searched on  and did not find lib32bz2-1.0 for Ubuntu 16.04. So how do i get this package which is available for trusty and precise? Resolving ambiguous overload on function pointer and std::function for a lambda using + I have terrible screen tearing, and I don't know how to go about trouble shooting it. I would LOVE to switch to Ubuntu completely, but with this tearing, it's just too irritating. It's only present on Ubuntu based distros, (including element OS). On Manjaro (Arch fork), it does not suffer from tearing.  PC Configuration:  Motherboard: Aorus Pro Wifi z390 CPU: Intel i7 9700k RAM: 32GB  GPU: Nvidia 1060 6GB DisplayPort output connects to monitor's HDMI  Monitor reported refresh rate 75hz, Ubuntu monitor settings reported refresh rate 74.97 (can change to 60hz, doesn't help). I followed Mike's solution from my first question:    My next question is how to add entries (for the abstract and acknowledgements pages) into the table of contents. Start a git commit message with a hashmark (#) I'm not that into dynamic programming languages but I've written my fair share of JavaScript code. I never really got my head around this prototype-based programming, does any one know how this works? var obj = new Object(); obj.prototype.test = function() { alert('Hello?'); }; var obj2 = new obj(); obj2.test();  I remember a lot discussion I had with people a while back (I'm not exactly sure what I'm doing) but as I understand it, there's no concept of a class. It's just an object, and instances of those objects are clones of the original, right? But what is the exact purpose of this &quot;.prototype&quot; property in JavaScript? How does it relate to instantiating objects? Update: correct way var obj = new Object(); // not a functional object obj.prototype.test = function() { alert('Hello?'); }; // this is wrong!  function MyObject() {} // a first class functional object MyObject.prototype.test = function() { alert('OK'); } // OK  Also these  really helped a lot. How to make password-less login work I generated a key and copied it to my target:  ssh-keygen ssh-copy-id username@hostname   I left the the location of the key as default and did not set any passphrase.  For a small while after that, I was able to ssh username@hostname without being prompted for a password, and then it stopped working for some reason. Consider a rocket in deep space with no external forces. Using the formula for linear kinetic energy $$\text{KE} = mv^2/2$$ we find that adding $100\ \text{m/s}$ while initially travelling at $1000\ \text{m/s}$ will add a great deal more energy to the ship than adding $100 \ \text{m/s}$ while initially at rest: $$(1100^2 - 1000^2) \frac{m}{2} \gg (100^2) \frac{m}{2}.$$ In both cases, the $\Delta v$ is the same, and is dependent on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this $\Delta v$. So I'd wager the same quantity of chemical energy is converted to kinetic energy, yet I'm left with this seemingly unexplained $200,000\ \text{J/kg}$ more energy, and I'm clueless as to where it could have come from. What can I expect from a Skype interview for PhD admissions?