In lecture we implicitly used the following theorem:  Let $R \subseteq S$ be two commutative rings and let $f \in R[x]$. If $f$ factors in $S[x]$ as $f(x) =g(x)h(x)$ with $g(x)$ a monic polynomial in $R[x]$, then $h$ is also in $R[x]$.  I do not understand why this is true. I assume that the assumption that $f$ is monic plays an important role here, but I do not see how I could use it. I tried to write out the product $$f(x) \cdot g(x) = (x^n+\ldots+a_0) \cdot (b_mx^m+\ldots+b_0)$$ and see if this would help, but to no avail. Could you please give me a hint?
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.
I don't see the statistics information that's supposed to be on the status bar running along the bottom of the main window towards the right in version 2.9. (Built-in) way in JavaScript to check if a string is a valid number Two envelope problem revisited I was thinking of this problem.  I believe the solution and I think I understand it, but if I take the following approach I'm completely confused. Problem 1:  I will offer you the following game. You pay me \$10 and I will flip a fair coin. Heads I give you \$5 and Tails I give you \$20. The expectation is \$12.5 so you will always play the game.  Problem 2:  I will give you an envelope with \$10, the envelope is open and you can check. I then show you another envelope, closed this time and tell you: This envelope either has \$5 or $20 in it with equal probability. Do you want to swap? I feel this is exactly the same as problem 1, you forgo \$10 for a \$5 or a \$20, so again you will always switch.  Problem 3:  I do the same as above but close the envelopes. So you don't know there are $10 but some amount X. I tell you the other envelope has double or half. Now if you follow the same logic you want  to switch. This is the envelope paradox.  What changed when I closed the envelope?? EDIT: Some have argued that problem 3 is not the envelope problem and I'm going to try and provide below why I think it is by analysing how each views the game. Also, it gives a better set up for the game. Providing some clarification for Problem 3: From the perspective of the person organising the game:  I hold 2 envelopes. In one I put \$10 close it and give it to the player. I then tell him, I have one more envelope that has either double or half the amount of the envelope I just gave you. Do you want to switch? I then proceed to flip a fair coin and Heads I put \$5 in and Tails I put \$20. And hand him the envelope. I then ask him. The envelope you just gave me has twice or half the amount of the envelope you are holding. Do you want to switch?  From the perspective of the player:  I am given an envelope and told there is another envelope that has double or half the amount with equal probability. Do I want to switch. I think sure I have $X$, hence $\frac{1}{2}(\frac{1}{2}X + 2X) &gt; X$ so I want to switch. I get the envelope and all of a sudden I am facing the exact same situation. I want to switch again as the other envelope has either double or half the amount. Analog voltage level conversion (level shift) I have a joystick that puts out analog voltage somewhere between 1.5V and 3.5V.  I want to transform this range of voltage to control a motor controller that takes analog voltage between 0V and 5V.  How do I level shift and amplify properly? How should I handle questions which are answered in the comments? Sometimes I have asked a question which got no answers, but is answered in the comments to the question.  How should I handle these questions? I can't accept an answer, so it is never an accepted question.  For example this one:   (ok here the answer in the comments is from me, but nevertheless I also have others of them) In The Last Jedi:     The Resistance fleet is tracked through hyperspace by the First Order after they make their jump away from their old base.   As soon as the FO ships appear, the Resistance immediately come to the (correct) conclusion as to the tech that was used.  Why would they assume the use of this tech, that they have never encountered before, as opposed to the more common technique of     tracking a ship through the use of a homing beacon. Let $M$ be a smooth manifold and let $N$ be a manifold with boundary. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}M $. Let $M$ be a smooth manifold and let $N$ be a manifold with boundary, both with the same dimension $n$. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}N  $.  I am trying to prove this theorem to prove a result about smooth embeddings.  Here is how I am thinking the problem could be solved. Assume thet $F(p)$ is a boundary point of $N$. Then there exists a chart $(V,\psi)$ at $F(p)$ such that $\psi(V) $ is an open subset of the upper half space $\mathbb{H}^{n}$. I guess we have to use some fact about $M$ having no boundary and the the fact that $dF_p$ is an isomorphism to show that there is a contradiction, but I cannot figure that out. python: restarting a loop The following definition is needed for my actual question:  For $A, B \in \mathcal{M}_{n \times n}$ define $$  \langle A, B \rangle = \text{Trace}(B^{T}A) = \sum_{j=1}^{n}\sum_{i=1}^{n}b_{ij} \, a_{ij} $$  And here is the actual question:  For $B \in \mathcal{M}_{n \times n}$ show that $$ \text{Trace}(B)^{2} \:\leq\: n \,\text{Trace}(B^{T}B) $$  I know I have to use the Cauchy-Schwarz Inequality, but I wasn't exactly sure how to.  Here is my attempt:  Since I only have one matrix I decided to use the identity matrix $$ \begin{align} \text{Trace}(B)^{2} \:&amp;\leq\: n \,\text{Trace}(B^{T}B) \\ \text{Trace}(I^{T}B)^{2} \:&amp;\leq\: n \, \langle B, B \rangle \\ \langle I_{n}, B \rangle^{2} \:&amp;\leq\: n \, \Vert B \Vert^{2} \\ \langle I_{n}, B \rangle \:&amp;\leq\: \sqrt{n} \, \Vert B \Vert \end{align} $$  I feel like I either did it incorrect, or that I'm close but am missing something. What is the relation between the Higgs field and chirality? Wikipedia  that the spontaneous breaking of chiral symmetry "is responsible for the bulk of the mass (over 99%) of the nucleons".  How do the nucleons gain mass from the spontaneous breaking of chiral symmetry? Why don't leptons gains mass from it? What is the role of the Higgs field in this all? Cannot install vim I am running Ubuntu 11.10, as Xubuntu, in an AMD 64 pc, configured as dual boot with Windows 7.  I tried installing vim with this:  sudo apt-get install vim   I got this error message:  The following packages have unmet dependencies:  vim : Depends: vim-common (= 2:7.3.154+hg~74503f6ee649-2ubuntu2) but 2:7.3.154+hg~74503f6ee649-2ubuntu3 is to be installed E: Unable to correct problems, you have held broken packages.   I only have a very faint idea of what the problem is, and none whatsoever on how to correct this. Hope you guys can help. What is wrong with this fake proof $e^i = 1$? It is said in many textbooks that alpha decay involves emitting alpha particles, which are very stable. Indeed, the binding energy () is higher than for $Z$-neighboring stable isotopes. But the binding energy is lower than, for example, ${}^9\mathrm{Be}$ (). My question is why aren't other nuclear compounds ejected from heavy nuclei, e.g. ${}^9\mathrm{Be}$?  The   $$e^{-\frac{4\pi}{\hbar}\frac{Ze^2}{4\pi\epsilon_0}\frac{1}{v_\alpha}}$$ decreases exponentially in $Z$, so it explains intuitively why lower-$Z$ particles would tunnel more often. Specifically, it would explain why we would see ${}^9\mathrm{Be}$ emission ${e^{-2}}\simeq 0.14$ times as often compared to ${}^4\mathrm{He}$ emission. Also, the particles need to form in the nucleus prior emissions; but with a similar binding energy per nucleon (~7.08 MeV for ${}^4\mathrm{He}$ vs. 6.47 MeV for ${}^9\mathrm{Be}$) and higher total binding energy for the ${}^9\mathrm{Be}$ nucleus, I would expect that its formation in the same order of prevalence as the alpha particle (according to Ohanian, between 0.1 and 1 alpha particles are in existence at any moment in time).  Can anyone explain this? A reference to an article/textbook would be preferable.  EDIT Same goes for ${}^{16}\mathrm{O}$ which is also a double magic isotope, as 'anna v' pointed out. For it, the Gamow factor is smaller by $e^{-4}\simeq 0.02$, and emission should still be viable. Problem about a process with bins of balls This is my circuit. I need to find the power dissipated by Vlamp. So the lamp turns on if the Vlamp is more than 2V. When the lamp is on it has an internal resistance of 10 Ohms, and when it is off it acts like an open circuit.  How would I find the power dissipated? I know how to do it for Voltage sources and Resistors, but not elements like these.  One more thing, I already figured out that: Battery has open circuit voltage of 4V R1 = 5 Ohms R2 = 5 Ohms k = 0.2A/V When lamp on and Vlamp = 2V, Rbat = 10 Ohms   . Get Emacs to ignore contents of \Sexpr{} command in Sweave document to prevent incorrect $-based syntax highlighting How to access a usb flash drive from the terminal? I only have access to the command line. I need to backup my data (on the user folder) to a pen (USB DOK).    How can I mount a flash drive manually? What should be the copy command? Last night I took some night sky shots, and I've noticed that with long exposure times (15s, 30s) the time it took to store the images in the camera memory card (jpeg) took roughly as long?  The actual file size is fairly small at 1 to 3MB per file.  Any idea why this is? How to remap an existing keybinding in Zathura Alignment issue for the last modified information in the "Top Questions" page In the  (homepage), the asked/modified X mins ago text was left-aligned next to the tags. Previously the text was right-aligned. Seems some alignment issue introduced on that page recently. Screenshot for reference:      Configuration Version     Operating System Windows 10 Pro   Mozilla Firefox 89.0 (64-bit)   Google Chrome 91.0.4472.101 (Official Build) (64-bit)     Update: I see the same alignment issue in all other Stack Exchange site's &quot;Top Questions&quot; page too.