Prove $I_k=I_N$ for some $k \ge N$.
Every principal ideal domain satisfies ACCP.
We have various ways of invoking apex. Triggers, Inbound Webservices, @InvocableMethod,@RemoteAction and what not. Often we call some or other utility methods from these execution contexts. I have some generic utility method,that should behave differently from the context it is called?  I know about Trigger.isExecuting but it only tells if its running in trigger context or not.  I have a utility method that creates lead from Webservices, VF button click and via Lightning . Whenever we create a lead I want some special operations to be done via the source it was originated.(Its not just updating a simple field leadSource, I have some methods that creates some childs and update other records of other type) . I can achieve this by passing an extra parameter to utility method to tell about source, but i have then to edit hundred of other class that uses this utility method. If i could dynamically find out via which context this piece of code is running,then i would have to change only one class to satisfy my requirement.    Is there something in APEX that tells in what context the apex method was called? Let $A$ be an infinite set. For each $a \in A$, let $x_a$ be a non-negative number. Let the value of the series $\sum_{a \in A} x_a$ be the value of the supremum of its finite partial sums, i.e.:      $$\sum_{a \in A} x_a = \sup_{n \geq 0} \sup_{(a_1, \ldots , a_n) \subset A} \left(x_{a_1} + x_{a_1} + \ldots + x_{a_n}\right)$$   (Supremum is well defined even when its value is infinite)       If $\sum_{a \in A} {x_a}$ is finite, prove that the set $A' = \left\{{a \in A | a &gt; 0}\right\}$   is countable.   My guess is that we have to write $A$ as a union of finitely many sets. How do I run executable scripts in Nautilus? How can I drive 12v LEDs from my Arduino Nano? Background I am building a scale RC helicopter and ordered a bunch of 3mm LEDs to give it some nice navigation and strobe lights which I plan to drive with an Arduino Nano. Problem is that I goofed and didn't notice that the LEDs are 12v until they arrived.  I know that the Arduino Nano can accept 12V on pin 30, but I think the voltage output on any of the &quot;D&quot; or &quot;A&quot; pins is always 5v (correct??). Question If the voltage output of the Arduino pins is 5v, then how can I drive the 12v LEDs using the Arduino, or is it impossible? Suppose I'm given the mean and one quantile (e.g. the 20% quantile) of a random variable $x$, and I want to find the parameters $\alpha$ and $\beta$ of a Beta distribution that has the same mean and quantile.  Is there an efficient way to do it?    Using just the mean, I know that since $\bar{x} = \frac{\alpha}{\alpha+\beta}$, we have $\beta = \frac{\alpha}{\bar{x}} - \alpha$.  So we only really have one parameter to estimate.  But I'm unsure how to use the quantile information to take the next step.  Maybe there's something I can do with the Incomplete Beta when I know the ratio $\frac{\beta}{\alpha} = \frac{1-\bar{x}}{\bar{x}}$?  I have access to R myself, so I could use a numerical optimizer for this, but ideally I need a method that can be carried out in Excel in someone else's environment.  Excel does have BETA.DIST() and BETA.INV() functions available.  A look-up table would be fine, but a closed-form formula would be better if it's possible. Is it acceptable to change the vid column of a term programmatically in the database? Suppose that $a_1,a_2, \cdots, a_n$ are $n$ different integers. Then $[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$.   I've no idea why it is true. Thanks very much. If two coins are flipped and one gets head, what is the probability that both get head? so I am trying to show that $f(x)$ divides $g(x)$ for all polynomials $g(x)$ satisfying that $g(A)=0$ where $f(x)$ is the minimal polynomial of a square matrix $A$.  I know from my professor that $f(x)$ is the minimal polynomial of $A$ so $f(A)=0$ so then $g(A)=0$. Therefore, $f(x)$ divides $g(x)$, i.e. $g(x)=f(x)h(x)$ where $h(x)$ is some other polynomial.  My prof told us to solve this question we need to let $b(x)=gcd((f(x),g(x))$ and by using Bezout's Identity, we need to show that $b(A)=0$. I am just having a little trouble trying to prove this. I know that this identity says:     For two polynomials $f_1(x)$ and $f_2(x)$, there exist polynomials   $g_1(x)$ and $g_2(x)$ such that $g_1(x)f_1(x) + g_2(x)f_2(x) = &gt; gcd(f_1(x), f_2(x))$.   And the uniqueness of minimal polynomial theorem is:     For a square matrix $A$, if $f_1(A) = f_2(A) = 0$ for two polynomials   $f_1(x)$ and $f_2(x)$, then $g(A) = 0$ for $g(x) = gcd(f_1(x), &gt; f_2(x))$. So the minimal polynomial of $A$ is unique up to a scalar.   So since $f(x)$ is already the minimal polynomial, I know that the $deg(b(x))=deg(f(x))$. I know how to prove the Bezout identity using the Euclidean algorithm, I'm just not exactly sure how I would show all of this using the notation he gave us of $b(x)=gcd((f(x),g(x))$. Any help at all would be appreciated. In the season 8 episode finale of Game of Thrones the highest important lords met with Grey Worm to free      Tyrion and Jon. While it was pretty clear why they wanted to free Jon it was unclear to me why they also asked for Tyrion.   To make things even more strange, they asked about his opinion of a potential solution which becomes the actual final one in the end      (choose king and nominate someone).   I thought they (besides Sansa and Bran) don't like him and wanted to judge him? Or am I missing something? How to install Windows on GPT disk without UEFI BIOS? Is it possible for an x86 processor to match an ARM processor in terms of performance per watt? From my personal experience with my tablet, and from the benchmarks and articles I've read, it always seems ARM processors, as seen in virtually all mobile devices, deliver incredible performance for the amount of power they consume.  It really seems like ARM is inherently more power-efficient than x86.  To me, it looks like that this is because x86 is a  while ARM is a . However, Intel Atom processors deliver very good performance relative to their power consumption as well.  In addition, having studied computer architecture, it is my understanding that in modern processors, the complex x86  is internally translated into simpler  that use an internal instruction set that is closer to the simpler ARM or MIPS ISAs than x86 proper.  As such, the ISA as presented to the programmer is little more than an interface to issue commands to the processor, rather than a representation of the actual low-level operations the chip performs.  Then again, implementing this translation layer requires additional silicon space on the chip...  That said, assuming that they are implemented using the same semiconductor process, is ARM inherently more efficient than x86? With a properly designed , is it possible for an x86 processor to deliver the same performance per watt as an ARM processor?  Remember, I'm looking for a well-written technical answer in the spirit of Stack Exchange and not mere speculation. Should your PhD students call you by your first name? Why don’t I have the Get Windows 10 icon? I want to get Windows 10, but I do not see this icon in the notification area (tray):    How do I get the icon to get Windows 10? Why I can't scan the WhatsApp QR? Why I can't scan the Whatsapp QR?   I reinstalled WhatsApp  I already have the WhatsApp Web Option  I put it to scan but it just doesn't scan it. I Installed a QR Scanner app and it works perfectly. I uninstalled the QR Scanner app to see if that was the issue and still doesn't work.   My phone is a Samsung Galaxy Ace. When typesetting URLs in LaTeX using the url package, they are set in a mono space font by default. While I just accepted that as a fact for a long time, I am now starting to wonder why?  In Browsers, for example, the URL in the address bar is also not monospace. It does not make the link more readable, in my opinion rather on the contrary. The only explanation I can come up with is that it sets the link apart from the remeaining text. But honestly, even that sometimes looks a bit obstrusive.  So is the reasons really just to set links apart from the text? Completeness and Incompleteness The identity cannot be a commutator in a Banach algebra? The Wikipedia article on  claims, without a proof or reference, that there does not exist a (unital) Banach algebra $B$ and elements $x, y \in B$ such that $xy - yx = 1$. This is surprising to me, but maybe the proof is straightforward; anyone have a proof and/or a reference?   More generally, I would have naively thought that I could embed any ring into a Banach algebra. I guess there are actually serious restrictions to doing this; are these issues discussed anywhere? Given the functional dependencies of a relation is there more then one way to normalize it? I have multiple Ubuntu machines at home and a pretty slow internet connection, and sometimes multiple machines need to be updated at once (especially during new Ubuntu releases.)  Is there a way where only one of my machines needs to download the packages, and the other machines can use the first machine to get the debs?  Does it involve setting up my own local mirror?  Or a proxy server?  Or can it be made simpler?