Running 12-2 Romex cable from the electrical panel to a J-box located in attic. J-box will connect two GFCI outlets and extend a 14-2 cable to the ceiling light fixture box, where other 14-2 cable will be used for a switch.   Circuit breaker is 20A.  Is that OK? Is it code compliance?
My refrigerator line is 20 amp with 12 gauge wire. Can I tap into that with 14/2 wire, and run a light switch and 1 LED light?
How to change the spacing between figures/tables and text? I was wondering if there is any way to lessen the spaces between figures and text. In my paper I have a lot of figures (graphics) and it is inconvenient for me to have default (large) spaces between text and graphics. Is there any command I could use to change the default settings? Why do we say that irreducible representation of Poincare group represents the one-particle state? Only because   Rep is unitary, so saves positive-definite norm (for possibility density), Casimir operators of the group have eigenvalues $m^{2}$ and $m^2s(s + 1)$, so characterizes mass and spin, and It is the representation of the global group of relativistic symmetry,    yes? OneDrive for Windows 10 - Remove logged out account Single word that means "to look down on others"? "Abusing" bounties for unlimited profit Six days ago I put a +50 bounty on  which wasn't asked by me, but to which I'd previously posted a pretty decent answer. Since then I've earned 130 reputation from upvotes on my answer, netting me +80 reputation and a bronze badge from no work.  It seems possible to slowly generate free reputation by "investing" in your answers like this. I expect I could repeat this perpetually and continue to profit from it.  This strikes me as an abuse of the bounty system. Should anything be done to prevent it?  EDIT: For more concrete questions:   If I put another bounty on this question after this one expired, would that be considered abuse? What if I did it several times? ( the answer was  "yes", but the author was doing so to promote his own project.) What if I didn't do it to the same question, but was constantly promoting questions I'd answered, in order to promote my answers? $f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $ Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0) = 0$ and $f'(x) = [f(x)]^{2}$, $\forall x \in \mathbb{R}$. Show that $f(x) = 0$,  $\forall x \in \mathbb{R} $.  I first (unsuccessfully) tried using the Mean Value Theorem, but this is in the Integrals chapter so the solution probably involves them. Can't really see where integrals come in here though.  What I've got so far:   (i) Since $f$ is differentiable, thus it is continuous and, hence,  integrable. Therefore $f^2$ is also integrable and as $f'=f^2$, $f'$ is too.   (ii) $f' \geq 0$, $\forall x \in \mathbb{R}$ Biasing circuit - Voltage follower circuit Using TIPA with fontspec How to intuitively understand eigenvalue and eigenvector? I'm learning multivariate analysis and I have learnt linear algebra for two semester when I was a freshman.  Eigenvalue and eigenvector is easy to calculate and the concept is not difficult to understand.I found that there are many application of eigenvalue and eigenvector in multivariate analysis. For example     In principal components, proportion of total population variance due   to kth principal component equal    $$\frac{\lambda_k}{\lambda_1+\lambda_2+...\lambda_k}$$   I think eigenvalue product corresponding eigenvector has same effect as the matrix product eigenvector geometrically.   I think my former understanding may be too naive so that I cannot find the link between eigenvalue and its application in principal components and others.   I know how to induce almost every step form the assumption to the result mathematically. I'd like to know how to intuitively or geometrically understand eigenvalue and eigenvector in the context of multivariate analysis(in linear algebra is also good).  Thank you! Isomorphism between two models of Hilbert's Axioms with different notions of congruence I've been reading about Hilbert's Axioms of geometry, and I have two models on $\mathbb{R}^2$ which satisfy the usual axioms of incidence, betweenness, and congruence.   The first model has a distance function given by  $$ d(A,B)=|a_1-b_1|+|a_2-b_2| $$ where $A=(a_1,a_2)$, $B=(b_1,b_2)$, and two segments $AB\cong CD$ if $d(A,B)=d(C,D)$.  The second model has a distance function given by $$ d(A,B)=\sup\{|a_1-b_1|,|a_2-b_2|\}, $$ and again $AB\cong CD$ if $d(A,B)=d(C,D)$.  The author claims that these two models are isomorphic in that there is a bijection between the sets of points in $\mathbb{R}^2$ that preserves lines, betweenness and congruence. I've been toying with it for a while, but it seems to me that the notions of congruence in particular are too different.   For example, in the first model, if $A=(0,0)$ and $B=(2,1)$, and $C=(2,0)$ and $D=(5/2,5/2)$, then $AB\cong CD$ since both segments have length $3$. To preserve congruence, I suppose two segments are congruent if the maximum of the displacement along the $x$-axis or $y$-axis by traveling along one segment is equal to the maximum of the displacement along either axis when traveling along the other segment. In this sense, how could you shape the segment $CD$ so that the maximum of the displacements along either axis is the same as the maximum of the displacement for whatever the image of $AB$ is?  In the first model, $AB$ is congruent to any segment which is the hypotenuse of triangle with legs that sum to $3$, and this sum could split into any two terms. I just don't see a function that could handle all these possibilities at once. Does anyone know of a particular isomorphism between these two? Thanks. JavaScript hard refresh of current page How to delay a program such as cairo-dock (No openGL) or Firefox?  Sometimes programs crash when started with other startup apps. I want programs to start 10 seconds late to avoid a crash. Baker-Hausdorff Lemma from Sakurai's book I'd like to show that, given to hermitian operators $A,G$ on a Hilbert space $\mathscr{H}$, the following identity holds: $$ e^{iG\lambda}A e^{-iG\lambda} = A + i\lambda [G,A] + \frac{\left(i\lambda\right)^2}{2!}[G,[G,A]]+\ldots+\frac{(i\lambda)^n}{n!}\underbrace{[G,[G,[G,\ldots[G}_{n\ times},A]]]\ldots]+\ldots $$ where $\lambda$ denotes a real parameter and $[\ \!,]$ indicates the commutator.  This is a proof left to the reader by Sakurai in his books on Modern Quantum Mechanics. List of Ubuntu Versions with Corresponding Linux Kernel Version Is there a list of Ubuntu versions with default corresponding Linux kernel versions somewhere?  I would specifically like to know the most recent version of Ubuntu that still used Linux Kernel 2.x. Unit ball in $C[0,1]$ not sequentially compact This question is taken from Saxe K -Beginning Functional Analysis.   Show that the closed unit ball in $C[0,1]$ is not compact by proving that it is not sequentially compact.   (It's assumed that we are using the uniform norm).  I've been working on this for ages but I could not come up with any sequence $\{f_n\}$ in the unit ball such that there exists $N\in \mathbb{N}$ such that for all $m,n\geq N$ we have that $d(f_n,f_m)&gt;c$. Should be a nice example of this, please help me! A large number of variants of this question were already asked here, including these - , , which are close, but none seem to answer my question. Assume that $n$ balls are thrown randomly and independently into $k$ bins.  What is the probability of finding $x$ empty bins? What is the expectation of the number of empty bins? Converting date to specified format and two date comparisions Is there a nice example of integral domain $R$ and $s,t \in R$, with $Rs \cap Rt$ not principal but lcm($s,t$) exists? I am currently searching for an integral domain $R$ which allows lcm, but where the intersection of two principal ideals is not necessarily a principal ideal (or at least an example with two certain elements).   As converse example (where the sum of two ideals is not principal but $R$ allows gcd) is $\mathbb Z[X]$. It allows gcd, but obviously $\left&lt;X+1,2\right&gt;$ is not principal.  An example of an integral domain where the intersection of two principal ideals is not principal would be $\mathbb Z[\sqrt{-5}]$ with the ideal $\left&lt;2\right&gt; \cap \left&lt;1+\sqrt{-5}\right&gt;$, but this doesn't allow lcm.  Since I can't find a nice example, I put the question here. Maybe someone has already seen this? (I am not completely sure, one even exists, but pretty certain there is an example. In the case my hunch is wrong, I would be happy, if someone could proof me wrong.) I was going to ask , but I came to the conclusion that I'm not going to do it (asking and building).  Now my question is: why is it (usually, if we leave the silly high-end prices alone) more expensive to build a bike from scratch? Traditionally  at the end of a filename after uploading it via the image uploader. That doesn't seem to work any more. I've tested it on Q&amp;A as well as chat.        (Yes, I know big M doesn't work)  Could we have this fixed please?