How do I prove that the standard definition of prime ideal is equivalent to that of Krull's?
Equivalence of definitions of prime ideal in commutative ring
A scene I made is not rendering, at all. This grey screen stays the same and nothing changes.  Here are my settings:  [ Prove $2^b-1$ does not divide $2^a + 1$ for $a,b>2$ I'm trying to prove $2^b-1$ does not divide $2^a + 1$ for $a,b&gt;2$. Can someone give a hint in the right direction for this? I did not know that you could find an answer for that. However, I can only use Excel so far to do it.  How to calculate 3.6! by hand? Why does it take so long for a computer monitor to change resolution? Why does it take 2-3 seconds rather than, say, 50 ms?  E.g. my current Windows 7 with strong GPU and 24 inch monitor takes 2-3 seconds. Why so long? I know some might say "it's JUST 2 seconds", but come on... Games produce hundreds of FPS, and monitors work at 60 Hz, so what is it that takes so much time?  I remember this was the same in just about any machine I ever worked with. Recently, there was some news that said that the researchers at NASA have come across some impossible kind of space engine which does not require any fuel. I have read at a few places like ,  and . They all say that this engine seems to violate the laws of physics including conservation of momentum.  The original paper is .  Is the statement that it violates conservation of momentum true? How do I avoid rendering glitches from overlapping faces? Style every four posts differently Possible Duplicate:          Is it possible to style every four posts differently? I mean something like this:  &lt;div class="row"&gt; &lt;div&gt;1st post&lt;/div&gt; &lt;div&gt;2nd post&lt;/div&gt; &lt;div&gt;3rd post&lt;/div&gt; &lt;/div&gt; &lt;div class="row&gt; &lt;div&gt;4th post&lt;/div&gt; &lt;/div&gt; ...Repeat   I'm using , so it would be great if I could experiment more with that.. :) Which characters are valid in CSS class names/selectors? What characters/symbols are allowed within the CSS class selectors? I know that the following characters are invalid, but what characters are valid?  ~ ! @ $ % ^ &amp; * ( ) + = , . / ' ; : " ? &gt; &lt; [ ] \ { } | ` # I have noticed that such terms as model hyperparameter and model parameter have been used interchangeably on the web without prior clarification. I think this is incorrect and needs explanation. Consider a machine learning model, an SVM/NN/NB based classificator or image recognizer, just anything that first springs to mind.   What are the hyperparameters and parameters of the model? Give your examples please. Intuitive explanation of entropy I have bumped many times into entropy, but it has never been clear for me why we use this formula:  If $X$ is random variable then its entropy is:  $$H(X) = -\displaystyle\sum_{x} p(x)\log p(x).$$  Why are we using this formula? Where did this formula come from? I'm looking for the intuition. Is it because this function just happens to have some good analytical and practical properties? Is it just because it works? Where did Shannon get this from? Did he sit under a tree and entropy fell to his head like the apple did for Newton? How do you interpret this quantity in the real physical world? Double value returns 0 Which email was the verification email sent to? I started my app today and it said that I needed to verify my email. But I don't know the email I chose. I logged on every email in but there was never a verify email. Is there a way to know your email on Pokemon Go? Why are my converted CMYK blacks coming out grey? I'm having a rather annoying problem! I have RGB images made in Photoshop that I need to convert to CMYK.  When I save the file my black background is converted to a light grey! I tried deleting my background and adding a pure black background in my PSD file but even that comes out grey! There is no way to save black? The colors all look fine in Photoshop, it's just when I save them as TIFF/JPEG.  How can I prevent this? Here is my script:  #!/bin/bash  echo "Digite o local em que deseja instalar o IGRAFU(pressione enter para  instalar em  ${HOME}/IGRAFO):" read caminho if test -z $caminho then caminho="${HOME}/IGRAFO" fi echo "O IGRAFU será instalado no diretório: $caminho" mkdir -pv $caminho  mv -v ./* $caminho echo "Pronto!"   At 'read caminho' I may receive from the user a path like ~/somefolder. When the script receives that kind of path both mv and mkdir won't make tilde expansion, so it will try to create a ~/somefolder and not /home/username/somefolder and therefore fail.  How do I ensure that the tilde will be converted into the HOME variable? I'd like to create a popup list on one of my Visualforce pages, for a user to select from a list of OrgWideEmailAddress records.  My goal is to limit the list to only those records that they're allowed to use.  I'm going to save the selection for use in later automatic emails.  Based on what I've read, I won't be able to send emails to certain OrgWideEmailAddresses:   The address hasn't been verified yet.  I don't see any fields in the schema that appear to show verification state. The address is only available to profiles that the user isn't a member of.  I don't see where the profile list is stored for each record.   Some of my problem could just be due to not understanding the standard objects so well, and perhaps there's a way to get at this through UserInfo, or by making my query on the User object which may have a child relationship to OrgWideEmailAddress since it's listed in the schema viewer of the Force.com IDE.  Here's the test code I'm working with:  for (OrgWideEmailAddress addr : [SELECT DisplayName, Address          FROM OrgWideEmailAddress ORDER BY DisplayName]) {     System.debug( addr.DisplayName + ' &lt;' + addr.Address + '&gt;'); }   I tried creating a new Messaging.SingleEmailMessage and then calling setOrgWideEmailAddressId() on it with each OrgWideEmailAddress Id, in hopes that invalid Ids would throw an exception, but to no avail.  My goal is to modify the test code to provide just the subset of records that the current user will be able to use. Cooperative game vs Multiplayer game Can any one tell what is difference between cooperative game and multiplayer game? I accidentally removed some files on a NTFS partition  How can I recover them staying inside Linux without rebooting ? How do I copy a property from an active object to selected objects? I need help to calculate the  Fourier transform of this funcion  $$b(x) =\frac{1}{x^2 +a^2}\,,\qquad a &gt; 0$$ Thanks. Where is the dimensions option?