I need to either prove this is an integral domain or give a counterexample.  I have yet to find a counterexample, but can't seem to prove it on my own.  Would love any advice/help! Thank you!
I have a doubt on something. I have a theorem that says that if $R$ is a ring, then $R/I$ is a domain iff $I$ is prime. Since $X^2-1$ is note prime, $\mathbb Z[X]/(X^2-1)$ is not a domain. But, $X^2-1=(X-1)(X+1)$, and since $X-1$ and $X+1$ are prime in $\mathbb Z[X]$, there are coprime, and thus, by the chinese remainder $$\mathbb Z[X]/(X^2-1)\cong \mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1),$$ since $\mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1)$ is a domain, then so is $\mathbb Z[X/(X^2-1)$. What is the mistakes here ?
nmake fatal error U1034: syntax error : separator missing I have to prove that a function $f:]0,1] \rightarrow \Bbb R$ : $$ f(x) = \begin{cases} \frac1q,  &amp; \text{if $x \in \Bbb Q$  with $ x=\frac{p}q$ for $p,q \in \Bbb N$ coprime} \\ 0, &amp; \text{if $x \notin \Bbb Q $} \end{cases} $$  is discontinuous in every point $x \in \ ]0,1] \cap\Bbb Q$.  And then to consider $x \in \ ]0,1] \backslash \Bbb Q$ and prove that it is continuous.  For now I learned different ways to prove continuity (epsilon-delta, sequences), but I'm never sure what would be better to use in each different case.  I wanted to prove the discontinuity by using sequences: $$\forall x_n \quad x_n\rightarrow a \quad \Rightarrow \quad f(x_n) \rightarrow f(a)$$  I tried creating a sequence $ x_n=\frac1n + a$, we know it converges to $a$ but $f(x_n)\ $ doesn't  converges to $\ f(a)$ because there would still be some points not in our set (irrational numbers that creates gaps).  But I don't think it works, so I'm asking you if you could help me solving the two questions. Can I get UAC prompt for user from batch file? Can I get UAC prompt for user from batch file?  For example I need put some keys to registry under CurrentControlSet from batch file by     reg import path\to\my.reg   PS. runas is not a solution as my customers uses different Windows localisation and I can not guess which name used for Administrator account (this can be Администратор)...  And script don't know administrator password.  And UAC dialog box does not require password entering... Get lists of dependent picklist options in Apex I have a pair of dependent picklists.  In Apex code, how can I determine what options are valid in the dependent field for each option in the controlling field?  I've tried using getPicklistValues(), but there doesn't seem to be any way of getting dependency information.  Example  Controlling_Field__c  Fruit, Vegetable, Dairy  Dependent_Field__c  (Fruit): Apple, Banana, Pear  (Vegetable): Tomato, Egglant, Lettuce  (Dairy): Milk, Cheese, Yogurt  What I'm Looking For  (In pseudocode, something like:)  controllingOptions = dependentfield.getPicklistValues(); for (option : controllingOptions) {     dependentOptions = dependentfield.getPicklistValuesFor(option);     do_something_with_it(); } I would like to clarify the differences between by value and by reference.  I drew a picture     So, for passing by value,  a copy of an identical object is created with a different reference, and the local variable is assigned the new reference, so to point to the new copy  How to understand the words: " If the function modifies that value, the modifications appear also within the scope of the calling function for both passing by value and by reference "  Thanks! Sponsor Gallery, Page, or Something Like That? nhj asked the other day,  In , I wrote:     We were thinking of putting together a "sponsor gallery" of sorts that listed out all of the sponsors with a brief description. Kinda in the vein of what we have on the every-now-and-then  at TDWTF.   So, consider this the discussion for that topic.   What would you like to see in a sponsor gallery? Should it be one big, long list? How does the order get determined? Should we organize the sponsors based on tags (and if so, what about sponsors who target like java and .net)? What about former sponsors... do they get removed from the list, or go on like the "past sponsors" section/page?  Keep in mind that this is not something we intend to charge extra for... this is more a service to the you, the members, than anything else. I've gotten postive feedback from the TDWTF "gallery", as a lot of readers come through RSS, have adblock running, etc. Hardware issue on iMac, flashing question mark I have an iMac 2008/2009 issue and recently the computer would not boot and I would get a flashing question mark. Ran through all the usual steps, try boot in safe mode, flash RAM,  try single user for fdisk, put in OS disk and boot from CD, etc... No dice.   Replaced the hard drive and put the OS disc in the system, press and hold option. (holding C didn't work) I can see the OS disk but when I click on it the system will act like it is going to run and then freeze. The best I got was the OS X disc that changed the screen to dark grey and told be the system had to restart.  Purchased a usb mount for an external hard drive. Was able to mount the old hard drive to a laptop and run disk utility. Everything checked out.  At this point I am pretty sure it isn't the hard drive.   Any other hard ware items this can point too? My iMAC can also "see" the hard drive (and the recovery partition) when connected via the USB and I hold "option" on start up. But it crashes when trying to boot from either.  I am going to take out and re-install the RAM for giggles but I am not sure what else to do.   Ideas are welcome. how to prove the chain rule? I have just learnt about the chain rule but my book doesn't mention a proof on it. I tried to write a proof myself but can't write it. So can someone please tell me about the proof for the chain rule in elementary terms because I have just started learning calculus. I was trying to solve internet problems, and seem to have made things worse: Now when I start the computer it gets stuck on "Booting system without full network configuration".   Does anyone know how to make it unstuck?   Alternatively: It got messed up when I changed the contents of /etc/network/interfaces, so presumably would work again if I changed it back. How can I edit this file during boot?   I'm using Ubuntu 12.10. How to generate multiple maps where one highlighted attribute changes? I have a layer that contains counties within the state of Washington.    I need to build one map (jpg) for each county.  For that specific map all counties will show as light grey, but the specific county will be green, and the file name will be the county name. Our high-school teacher told us that the Earth pulls us with some force $F$ and we pull the Earth with the same force $F$. Within Newtonian physics this is true because of , but let’s consider . My mass is small; so I don’t warp space-time much. But Earth’s large mass warps space-time to a far greater extent.  So do I pull Earth with the same force it pulls me? If yes, how? force long text inside \item[{\texttt{}] to wrap I was trying to prove this inequality using induction, but couldn't do.  Question: Suppose $a$ and $b$ are real numbers with $0 &lt; b &lt; a$. Prove that if $n$ is a positive integer, then:  $$a^n-b^n \leq na^{n-1}(a-b)$$ I'm trying to convert some strings that are in French Canadian and basically, I'd like to be able to take out the French accent marks in the letters while keeping the letter. (E.g. convert é to e, so crème brûlée would become creme brulee)  What is the best method for achieving this? What, precisely, is a confidence interval? I've been working on the Android SDK platform, and it is a little unclear how to save an application's state. So given this minor re-tooling of the 'Hello, Android' example:  package com.android.hello;  import android.app.Activity; import android.os.Bundle; import android.widget.TextView;  public class HelloAndroid extends Activity {    private TextView mTextView = null;    /** Called when the activity is first created. */   @Override   public void onCreate(Bundle savedInstanceState) {     super.onCreate(savedInstanceState);      mTextView = new TextView(this);      if (savedInstanceState == null) {        mTextView.setText("Welcome to HelloAndroid!");     } else {        mTextView.setText("Welcome back.");     }      setContentView(mTextView);   } }   I thought it would be enough for the simplest case, but it always responds with the first message, no matter how I navigate away from the app.  I'm sure the solution is as simple as overriding onPause or something like that, but I've been poking away in the documentation for 30 minutes or so and haven't found anything obvious. Question: Can anyone shed some light on this formula? I can't find any information on it.  It has three corollaries that I also need to understand: Filtered Questions Asked By Accepted Answer/ No Accepted Answer I think it's helpful if we can go to a page and view all our own questions that   Does not have an accepted answer, and/or Does not have an answer yet.   The reason I asked for them is because, I sometimes need to review through all the questions, and pick those that don't have proper answers so that I can perform further operations on them ( such as offering a bounty, or accept an existing answer). How would one write object-oriented code in C? What are some ways to write object-oriented code in C? Especially with regard to polymorphism.    See also this Stack Overflow question . What is the difference between the GNU Makefile variable assignments =, ?=, := and +=? Can anybody give a clear explanation of how variable assignment really works in Makefiles.   What is the difference between :   VARIABLE = value  VARIABLE ?= value  VARIABLE := value  VARIABLE += value   I have read the  in GNU Make's manual, but it still doesn't make sense to me.