Metal-softening tool
Device that makes metal malleable
Let $R$ be a sub-ring of a ring $S$. Let $f,g$ be non-zero polynomials in $R[x]$ and assume that the leading coefficient of $f$ is a unit in $R$. If $f$ divides $g$ in $S[x]$, show that $f$ divides $g$ in $R[x]$.   Thoughts: If $f$ divides $g$ in $S[x]$, then  $g(x) = f(x)q(x)$ for some $q(x)$ in $S[x]$. From here I'm not sure what the question is asking. Do I have to show that $q(x)$ is an element of $R[x]$ ? Insights appreciated. byte[] to hex string How do I convert a byte[] to a string? Every time I attempt it, I get      System.Byte[]    instead of the value.  Also, how do I get the value in Hex instead of a decimal? I am considering trying to setup a  system. It seems like this would be pretty straight forward if I had two graphics cards or if I didn't need hardware based OpenGL acceleration. Is it possible to setup a multi seat system with only a single dual-headed graphics card? Do vulnerability and resistance cancel out? Why does no line appear when using Interpolate Line in ArcMap? Finding how multiplication and addition behave on $\mathbb{F}_4$ without any result I'm currently tutor in an undergraduate course and the students are asked to find how addition and multiplication behave on the field with 4 elements: $\{0,1,x,y\}$. In the solution the teacher gave me, he uses the fact that $(\mathbb{F}_4^*, \cdot )$ is cyclic with order 3 to find that $x$ and $y$ are both generators and that $(\mathbb{F}_4,+)$ is a group of order 4 to deduce that $0=1+1+1+1=(1+1)(1+1)$ and hence, $(1+1)=0$ since a field is integral.   However, I'm not satisfied with this reasoning because this question is asked to student in first year and they juste have been introduced to fields (and this was part of an analysis course so they won't see any result soon). So I tried to find how it behaves by myself, using as few results as possible.  Multiplication is the easy part, $0\cdot a$ and $1 \cdot a$ is trivial for all $a\in \mathbb{F}_4$. Then, $x\cdot y\not = 0$ because it is invertible, and it can't be equal to $x$ or $y$ because that would imply the other element to be equal to $1$. Hence, $x\cdot y=1$. We are left to find what is $x^2$ (and similarly, $y^2$). $x^2$ cannot be $0$, nor $x$. If $x^2=1$, then $y=(x^2)y=x(xy)=x\cdot 1=x$. Contradiction. Hence $x^2=y$ and by the same reasoning, we have $y^2=x$.  It was easy with the multiplication, but I can't figure out any non-trivial result concerning addition. My intuition tells me that distributivity should be used here but I'm feeling like it's just converting a problem into another one because we don't know anything about addition.  Does anyone here have an idea of how addition can be "found" without using any advanced result (or as few results as possible)?  Thanks in advance. How do I design a 2A or more power supply for my consumer USB devices? I understand that many of my consumer devices charged by USB ports can charge at higher rates than 0.5A.  However, in testing, I've found they don't consume more than this. Test Setup  I have a power supply that outputs between 0 and 30 volts, and 0 to 20 amps. I set the voltage to 5 volts, then connected up a female USB to the wires correctly. When I plug in a Samsung Galaxy S4 Mini, it pulls around 0.44 amps. It is the same with a Samsung Galaxy Tab Pro. However, when I connect an iPad Air with an official lightning cable, it only pulls around 0.11 amps. Also, I tried connecting a ZGPAX S28 smartwatch and it was still about 0.44-0.45 amps.  The power supply is more than capable of outputting the maximum 2.1 amps, so why doesn't it to at least the tablets? What do I need to do in my test setup to convince the devices to consume their maximum charging current? Some time ago, I baked a cake (Sachertorte to be exact). The taste and texture was fine, so my recipe is okay.  My biggest issue was the shape. In the middle it was really a lot higher than on the sides. I would like to redo it in the near future, but preferably with a flat(ter) surface. (Especially since I put icing on top and I don't want it to drip off this time.)  Anyone knows the cause of my problem, or even better a solution? I think I put enough butter on the side of my springform pan, in case this is an option. Why do we need to discard one dummy variable? Jquery load() and PHP variables Why is HMAC-SHA1 still considered secure? This Q &amp; A   says that the security of HMAC-SHA1 does not depend on resistance to collisions? Are they are saying specifically with respect to HOTP or for HMAC-SHA1 for any use?     If HMAC-SHA1 is used for verifying the integrity of a document and you can find a collision, you can make changes in the document, right?      Even the RFC says that resistance to collisions is relevant       The security of the message authentication mechanism presented here depends on cryptographic properties of the hash function H: the resistance to collision finding (limited to the case where the initial value is secret and random, and where the output of the function is not explicitly available to the attacker), and the message authentication property of the compression function of H when applied to single blocks    What am I missing here? How long was Luke Skywalker's training with Yoda? From the answer to the question  :     We also don't know how much time Luke spent training.   I tried to check some references like  and  related section on their Wookieepedia articles, but there is not even a clue of the amount of time he spent there.  Is there any canon information on the subject? Or any extrapolation of how long Luke's journey on Dagobah was? Was it a day? Several weeks? Some months? AES Mix Column Transformation Section 3.4.3 "The MixColumns Step"(page 39) of , it states     The columns of the state are considered as polynomials over   $\operatorname{GF}(2^8)$ and multiplied modulo $x^4 + 1$ with a fixed polynomial $c(x)$.      The polynomial $c(x)$ is given by   $c(x) = \mathtt{03} \cdot x^3 + \mathtt{01} \cdot x^2 + \mathtt{01} \cdot x + \mathtt{02}$      Let $b(x) \equiv c(x) \cdot a(x) \pmod{x^4 + 1}$. Then         How this polynomial multiplication modulo $x^4+1$ is converted to Matrix Multiplication? The screen of my phone is physically broken, so it doesn't display anything, but the touch aspect does seem to be working, as I can unlock my phone, and get haptic feedback.  Before I go to get it replaced, I'd like to back up some personal data from the built-in memory to the SD card. I'm fairly certain that if I can get adb to recognize the device, then I can use  to see where I am touching the screen and do what I need to do. Problem is, when I plug it in, it's not recognized, while my emulator is.  I suspect that I might not have USB debugging enabled, which I think is necessary for adb to work. If this is the case, can someone give me screenshots of their Droid X (running 2.3.X) going from the home screen, through all the menus, to activating debug mode? Hopefully I can do it by dead reckoning and adb from there.  Edit: I haven't changed anything about the ROM or launcher; it's all stock. Node access based on view mode When writing letter to an academic professor (not necessarily from student to professor), what is the appropriate way to address his/her title?  I have seen in letters using  Dear Prof. X even when the addressee is not a full professor  Dear Dr. X though, knowing that the addressee is a full professor  Dear Mr/Mrs X though, knowing that s/he has an academic title A function $f: \mathbb R \to \mathbb R$ satisfies equation $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb R $ How do I create a Java string from the contents of a file? Is there a way of finding a formula for $\sum\limits_{k=1}^n k^k$? Maybe I'm missing something really obvious, but I've looked around a bit on the Internet and I haven't been able to find anything.  So, what I'm looking for is a formula in closed form to generate the sequence $1,5,32,288,3413,\dots$ Why does the definition of limits of a function have strict inequality? Definition (As written in Michael Spivak's Calculus)     The function $f$ approaches a limit $l$ near $a$ means: for every $\epsilon &gt;0$ there is some $\delta &gt; 0$ such that, for all $x$, if $0&lt;|x-a|&lt;\delta$, then $|f(x)-l|&lt;\epsilon$.   my question is: why can't it be: $$0&lt;|x-a|\leq \delta,|f(x)-l|\leq \epsilon$$ After looking at limits of functions for a long time just to grasp it's meaning and using the definition quite a lot solving homework I realized I keep writing the same inequality without really understanding why.  The only explanation given in Spivak's book for this part of the definition goes over it without explaining the inequality. I tried looking for an explanation myself but wasn't really able to find anything wrong with it. Is it also possible to write the definition like that or is there a problem with that?   (first non-homework related question :p)