In $\mathbb{Z}\left [ x \right ]$, let $I = \left \{ f\left ( x \right ) \in \mathbb{Z}\left [ x \right ]: f\left ( 0 \right ) \text{ is an even integer} \right \}$   In fact, $I=\left \langle x,2 \right \rangle$  How do I show that the order of $\mathbb{Z}\left [ x \right ]/I$ is 2?
In $\mathbb{Z}\left [ x \right ]$, let $I = \left \{ f\left ( x \right ) \in \mathbb{Z}\left [ x \right ]: f\left ( 0 \right ) \text{ is an even integer} \right \}$. Is $I$ a Prime ideal of $\mathbb{Z}\left [ x \right ]$?  Let $f_{1}\left ( x \right ),f_{2}\left ( x \right ) \in \mathbb{Z}\left [ x \right ].$ Let $f_{1}\left ( x \right )f_{2}\left ( x \right ) \in I$. Clearly, I is a proper ideal of $\mathbb{Z}\left [ x \right ]$ since the elements in $\mathbb{Z}\left [ x \right ]$ evaluated at 0 can produce an odd integer but this cannot be true in I. Now, $f_{1}\left ( 0 \right )f_{2}\left ( 0 \right )$ is an even integer in I. Hence, either $f_{1}\left ( 0 \right )$ has an even constant integer or $f_{2}\left ( 0 \right )$ has an even constant integer. Thus, I is a prime ideal by definition. Is this correct? I would now like to show if I is a maximal ideal. Again, clearly, I is a proper ideal by the above argument. Let B be an ideal of $\mathbb{Z}\left [ x \right ]$ and I $\subseteq B\subseteq \mathbb{Z}\left [ x \right ]$. How should B be defined so that I would conclude either $B=I$ or $B=\mathbb{Z}[x]$?
How to establish this inequality: $(1-a)(1-b)(1-c) \geq 8abc$ for $a+b+c=1$? Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c = 1$. Then how to establish the following inequality? $$ (1-a)(1-b)(1-c) \geq 8abc.$$  My effort:   Since $a+b+c =1$, we can write $$  (1-a)(1-b)(1-c) = 1 - (a+b+c) +(ab+bc+ca) - abc = 1 - 1  +(ab+bc+ca) - abc =  (ab+bc+ca) - abc = abc(a^{-1} + b^{-1} + c^{-1} ) - abc = 3abc (\frac{a^{-1} + b^{-1} + c^{-1}}{3}) - abc.$$ What next? A. $\lim_{n\to\infty}a_nb_n=?$ given that $\lim_{n\to\infty}a_n=0$ and $ \lim_{n\to\infty}b_n=\infty$  B. $\lim_{n\to\infty}{a_n \over b_n}=?$ given that $\lim_{n\to\infty}a_n=0$ and $\lim_{n\to\infty}b_n=0$  C. $\lim_{n\to\infty}{a_n \over b_n}=?$ given that $\lim_{n\to\infty}a_n=\infty$ and $\lim_{n\to\infty}b_n= \infty$    These seem to be difficult situations I come across when I have to evaluate limits.   Some examples,    $\lim_{n\to\infty}n^a\sin(1/n^b)$ with $a,b&gt;1$   $\lim_{n\to\infty}{n^a\over \sin(1/n^b)}$ with $a,b&gt;1$   $\lim_{n\to\infty}{1/n^a\over \sin(1/n^b)}$ with $a,b&gt;1$   $\lim_{n\to\infty} nf(x_n)$ [where we only know that $\lim \ f(x_n) = 0]$    My first question is : Are the two situations  A and B  equivalent ? I feel that they are equivalent as $\lim_{n\to\infty}b_n=\infty \iff \lim_{n\to\infty}{1\over b_n}=0$, The problem is that I'm not sure if we can apply the product rule for limits to take the limit inside.  My second question is : What are some ways to solve such problems ? I only know that we can apply stolz- cesaro in some cases. I have a Canon XTI which I've been using for a few years now. Most of the time I'm been shooting in one of the "semi-manual" modes like aperture priority or shutter priority, but now I'd like to try using full manual mode. I should have a new EF-S 17-55 f/2.8 delivered when I get home today. Before I was using a variable aperture Sigma 17-70 which made manual mode more difficult with the aperture changing arbitrarily.  I understand the effect of changing aperture and shutter speed, but right now I'm accustomed  to only controlling one at a time. The camera is usually setting the other to get the right exposure automatically for me. So if I'm controlling both how do I know I have the correct exposure? I understand I need to use the metering in my camera but I'm not sure exactly how do do that.   First when should I be using each of the metering modes (Evaluative, Partial, Center weighted, or Spot)? These just change the area of the scene which is metered correct?  How do I know what the meter should read off a specific object? Obviously if I just happen to be shooting a 18% gray wall then it should be at zero, but that can't apply to everything? If I'm shooting a black cat and take a spot meter reading off of the cat then the meter should read less then zero right? But how do I know how much less then zero? Are there rules of thumb I can follow? I've ordered a couple gray cards with my new lens. So experimenting with those should help.  How often should I be changing these settings? Say I'm at a party indoors which is consistently lit. I should be able to set the aperture, shutter speed, and ISO and just forget about it unless the lighting changes correct? Does the same apply to outdoor scenes? I've been hesitant to use manual mode in the past because I felt like I was spending a lot of times messing with settings before each shot. But not I'm realizing that this is really only the case when the lighting changes, or I want a different style of photo such as a different depth of field.  Finally, does exposure change with distance to the subject? For example say I want to take a picture of a statue with correct exposure using a gray card. I'd place the card on the statue, and while being careful not the block the light on the card find the correct exposure by spot metering the card. If I were then to step back 20' to get the whole statue in frame would I still have the correct exposure? How do I create files using the lines in a .txt file? For example I have a file called fileNames.txt.  cat &gt;fileNames returns  abet  able  abacus   How do I use fileNames to create abet, able, and abacus as .txt files?  I tried using pipes with cat and touch. I want LaTeX not to give warnings for overfull hboxes up to some maximum of say 2 pts. How to achieve this? Is there any built-in software or terminal method allowing me to view the hardware profiles on my system? Windows equivalent of such a feature would be Device Manager. iPhone stuck on Connect to iTunes My iPhone is stuck on Connect to iTunes Screen (support.apple.com/iphone/restore iTunes logo and up arrow with lightning cable) I have Tried :-   Connecting it to iTunes and Updating, It updates Successfully.But on the same screen again   Restoring, It restores successfully.But on the same screen again  Tried Recovery Mode Screen Method and DFU method Tried Closing and Restarting iTunes Tried Deleting 'iPhone Software Update' folder in iTunes and Re-Downloding Updates    Nothing Works... Any Help will be appreciated caching JavaScript files Which is the best method to make the browser use cached versions of js files (from the serverside)? What's the significance of conditional and absolute convergence? I know how to find if a series is conditional or absolute convergence by taking the absolute value and seeing if that absolute sum converges (then the original sum is absolute). If the absolute diverges but the original converges, then that's conditional.   But my question is: is there a significance? Is absolute convergence series bigger than absolute? Does a series being absolute or conditional affect rate of convergence, or radius? Does it affect symmetry?   Looking at Google, I only found tests on how to find if it's absolute or conditional. But I want to find meaning or significance. Or at least more mathematical properties. Why every polynomial over the algebraic numbers $F$ splits over $F$? I read that if $F$ is the field of algebraic numbers over $\mathbb{Q}$, then every polynomial in $F[x]$ splits over $F$. That's awesome! Nevertheless, I don't fully understand why it is true. Can you throw some ideas about why this is true? Sorting an Array of Objects in PHP In a Specific Order I have two arrays in PHP. The first array ($author_array) is comprised of user_ids in a particular order, like so: (8, 1, 6)  The second array ($user_results) is comprised of an array of objects like so:  Array (     [0] =&gt; stdClass Object         (             [ID] =&gt; 1             [user_login] =&gt; user1         )     [1] =&gt; stdClass Object         (             [ID] =&gt; 6             [user_login] =&gt; user6         )     [2] =&gt; stdClass Object         (             [ID] =&gt; 8             [user_login] =&gt; user8         ) )   I'd like to "sort" the second array so it's in this order, which matches the order of the values in the first array of (8, 1, 6). So it'd look like this:  Array (     [0] =&gt; stdClass Object         (             [ID] =&gt; 8             [user_login] =&gt; user8         )     [1] =&gt; stdClass Object         (             [ID] =&gt; 1             [user_login] =&gt; user1         )     [2] =&gt; stdClass Object         (             [ID] =&gt; 6             [user_login] =&gt; user6         ) )   I'm weak on data structures. How could I do this? :-)  Thanks in advance for your help!  -Bob What is the volatile keyword useful for? At work today, I came across the volatile keyword in Java. Not being very familiar with it, I found . Given the detail in which that article explains the keyword in question, do you ever use it or could you ever see a case in which you could use this keyword in the correct manner? Is there a good way to remove HTML from a Java string? A simple regex like replaceAll(&quot;\\&lt;.*?&gt;&quot;, &quot;&quot;)   will work, but things like &amp;amp; wont be converted correctly and non-HTML between the two angle brackets will be removed (i.e. the .*? in the regex will disappear). What is a "loop device" when mounting? Confidence Interval Calculation I have temperature data of July 1991 to 2007, and wants to find the Confidence Interval at 95%, which distribution is best fit the data, Z or T. Please give me reason also.    minTemp = 14.4444, maxTemp = 38.3333, SizeofData = 527, (17 years) hist(July, 20), histfit(July, 20); Add new alias to Gmail without SMTP (forwarding-only address) I used to be able to go to Accounts and Import → Add another email address you own and add an email account that's only a forwarding address. That is, it's not a mailbox and I don't have SMTP access to it.  Gmail would send me a confirmation email to that address, I'd confirm it's mine, and done. Now it only gives me the option to add SMTP credentials.  Did they stop supporting the previous method? Is there any other way I can get that option back? Ubuntu has been crashing on me recently. I think its because it runs out of memory so I ran the free -m command and found that my memory usage was really high. So then, i ran top to find the culprit, but the displayed processes were using less than 1.5% of memory. How do I know which program is making ubuntu crash/run out of memory? Below is the output:  shafee@shafee-pc:~$ free -m              total       used       free     shared    buffers     cached Mem:          3824       3714        110          0        978       1954 -/+ buffers/cache:        780       3044 Swap:           99          0         99  shafee@shafee-pc:~$ top top - 02:12:14 up  1:24,  2 users,  load average: 0.16, 0.42, 1.49 Tasks: 182 total,   1 running, 181 sleeping,   0 stopped,   0 zombie Cpu(s):  2.9%us,  1.9%sy,  0.3%ni, 79.3%id, 15.5%wa,  0.0%hi,  0.2%si,  0.0%st Mem:   3916708k total,  3803848k used,   112860k free,  1002308k buffers Swap:   102396k total,        0k used,   102396k free,  2001852k cached    PID USER      PR  NI  VIRT  RES  SHR S %CPU %MEM    TIME+  COMMAND              4200 root      20   0  289m  53m  38m S    2  1.4   1:06.45 Xorg                 5590 shafee    20   0 19348 1368  956 R    2  0.0   0:00.01 top                     1 root      20   0 24124 2136 1264 S    0  0.1   0:02.05 init                    2 root      20   0     0    0    0 S    0  0.0   0:00.00 kthreadd                3 root      20   0     0    0    0 S    0  0.0   0:24.23 ksoftirqd/0             6 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/0             7 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/1             9 root      20   0     0    0    0 S    0  0.0   0:00.11 ksoftirqd/1            11 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/2            13 root      20   0     0    0    0 S    0  0.0   0:03.89 ksoftirqd/2            14 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/3            16 root      20   0     0    0    0 S    0  0.0   0:00.16 ksoftirqd/3            17 root       0 -20     0    0    0 S    0  0.0   0:00.00 cpuset                 18 root       0 -20     0    0    0 S    0  0.0   0:00.00 khelper                19 root       0 -20     0    0    0 S    0  0.0   0:00.00 netns                  21 root      20   0     0    0    0 S    0  0.0   0:00.01 sync_supers         shafee@shafee-pc:~$   Or is it normal to have 110mb of free memory and is my issue related to something else? How can I add a folder to the launcher for quick access to it. In this particular case I wish to drag it to the launcher. How to call a SOAP web service on Android I am having a lot of trouble finding good information on how to call a standard SOAP/WSDL web service with Android. All I've been able to find are either very convoluted documents and references to "kSoap2" and then some bit about parsing it all manually with . OK, that's fine, but it's 2008, so I figured there should be some good library for calling standard web services.  The web service is just basically one created in . I would like to have IDE support for generating the plumbing classes. I just need the easiest/most-elegant way to contact a  based web service from an Android-based phone. Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ if $m &lt; 0$. This statement came up in an algebraic geometry text with no explanation provided, and I'm trying to understand why it's true.   Thoughts so far:   Since $H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m))$ is the vector space of regular sections of $\mathcal{O}_{\mathbb{P}^n}(m)$ over $\mathbb{P}^n$, we can try to find a basis. Since the regular sections are the degree $m$ polynomials in $x_0, \dots, x_n$, it seems like the collection of degree $m$ monomials in $x_0, \dots, x_n$ should work. But I don't see how there are ${n + m \choose n}$ of these. Is it just a basic counting argument that I'm struggling with?  On a side note, I think I understand what global sections of $\mathcal{O}_{\mathbb{P}^n}(m)$ look like, but how can we describe the regular sections over a distinguished open set $D(f)$? Is it something like degree $m$ polynomials with denominator $f$?