Split Epimorphism as surjective function
Is this an equivalent formulation of "surjective" resp. "epimorphism"?
Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups? $AB-BA$ is a nilpotent matrix if it commutes with $A$ I saw this in a  and am putting it here for posterity.  Problem: Let $A$ and $B$ by square matrices and set $C=AB-BA$. If $AC=CA$, prove $C$ is nilpotent. Is there a word or phrase to define several words linked by hyphens, such as in "a sit-back-and-wait-for-it attitude" some more examples:   "And she gave me that aren't-I-just-gorgeous smile." "The I-did-it-my-way approach." "A from-this-day-forward-I-have-no-son scene." An inequality from Littlewood's Miscellany What's the recommended max width for the main container and for the main column, for large screens? Based on this question,  and its comments and answers, I did some research, but I didn't find a complete answer.   For me it is obvious that it is a good decision to limit the width of the main container, but what is the recommended max-width for the main container for large screens, in 2017, when the screens become larger and larger? What variables should be considered in this decision?  Also, let's say a website has multiple columns in the main container, what is the max-width for the main column?  Update: I have now a related question. I recommended to one client to reduce the width of the main container on the homepage, but I can't convince him. It is a website with photos, like pexels.com. It is in this case important to limit the width if the elements are not one more important than other? Low Voltage AC Smoothing Capacitor Calculation I have a 9V AC source which will have a max load of 100mA.  I'm trying to figure the value of a smoothing capacitor with less than a 10% ripple.  I don't have a scope or else I'd just try a bunch out.  I'm just a hobbyist, but I'm trying to understand this.  Since I'm not sure if I'm doing this right, I'll just step through what I've done so far.  First I found this formula: $$V_{ripple}={V_p \over R_LC} \times \Delta T$$  I'm using 60Hz 9VAC that is full wave rectified and I want a 10% ripple so:  $$(9V \times .1)={9V \over R_LC} \times {1 \over 60Hz}$$  I'm calculating my resistive load with Ohm's law. $$R = {9V \over .1A} \quad R = 90 \Omega$$  So plugging that in I get: $$.9V={9V \over 90\Omega \; C} \times {1 \over 60Hz}$$  If I solve for C (admittedly using wolfram alpha), I get: $$C={1\over 540}F \quad C=1851 \mu f$$  1850µf seems pretty high to me...  And if I lower the ripple percent it get crazy high.  Is this the right way to find this?  Or did I screw up somewhere?  Also, my circuit has the capacitor between the full bridge rectifier's positive and negative outputs.  I've only seen this type of smoothing circuit, but maybe my circuit needs to change? If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show that $\lim _{x\to \infty } (f(x+1)-f(x))=0$. Having some trouble proving this:     If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show: $\lim _{x\to \infty } (f(x+1)-f(x))=0$   Attempt: from $\int \lim _{x\rightarrow \infty } f^\prime(x) \mathrm{d}x=0$ we can say that $f^\prime(x)=const$ would we then have to show that  $\lim _{x\to \infty } (f(x+1))$ is also a constant? From there if we show that both are equal then the difference is zero. I am on the right track?  Thank you for the help. What is 'deictic time'? [5]     i  If she beats him he’ll claim she cheated.    [non-deictic past]     ii If you eat any more you’ll say you don’t want any tea.[non-deictic present]      The preterite and present tense inflections on cheat and do indicate that Tr is respectively anterior to and simultaneous with To, but here  To is clearly not Td. The time of the (possible) cheating is not anterior to the time of my uttering [5i], but to the time of his (possibly) making a claim of cheating.                       — The Cambridge Grammar of the English Language, p126   I can understand To (time of orientation = utterance time), yet don’t get what the ‘deictic time (Td)’ is. Would you let me know the Td? I'm fitting a multiple linear regression model between 4 categorical variables (with 4 levels each) and a numerical output. My dataset has 43 observations.  Regression gives me the following $p$-values from the $t$-test for every slope coefficient: $.15, .67, .27, .02$. Thus, the coefficient for the 4th predictor is significant at $\alpha = .05$ confidence level.  On the other hand, the regression gives me a $p$-value from an overall $F$-test of the null hypothesis that all my slope coefficients are equal to zero. For my dataset, this $p$-value is $.11$.  My question: how should I interpret these results? Which $p$-value should I use and why? Is the coefficient for the 4th variable significantly different from $0$ at the $\alpha = .05$ confidence level?  I've seen a related question, , but there was an opposite situation: high $t$-test $p$-values and low $F$-test $p$-value. Honestly, I don't quite understand why we would need an $F$-test in addition to a $t$-test to see if linear regression coefficients are significantly different from zero. I have a Python  object. What is the best way to subtract one day? Beamer: how to justify the whole text? How can I justify the text in the whole document in beamer without using \usepackage{ragged2e} and \justifying at the beginning of each frame?  \documentclass[8pt,handout,t]{beamer} \usepackage{pifont} %used inside itemize environment \item[\ding{226}] \usepackage{lipsum} \begin{document}  \begin{frame} \begin{columns}[T]  \begin{column}{0.3\textwidth} \includegraphics[width=\columnwidth]{example-image-1x1} \end{column} \begin{column}{0.65\textwidth} \begin{itemize} \item[\ding{226}] \lipsum[2] \item[\ding{226}] \lipsum[3] \end{itemize} \end{column}  \end{columns} \end{frame}  \begin{frame}  \lipsum[3]  \begin{itemize}    \item[\ding{226}] \lipsum[4]  \end{itemize} \end{frame} \begin{frame}   \begin{columns}[t] \begin{column}{0.4\textwidth} \lipsum[3] \end{column} \begin{column}{0.4\textwidth} \lipsum[5] \end{column} \end{columns} \end{frame} \end{document}     Edit: I had already checked  out but it does not work for both the columns and the itemize environment. How can I take a photo with everything in focus with my DSLR? Where can I find the status of the released versions of Ubuntu? Existence of elements of order $p^k$ in a finite group By Cauchy's theorem, if a prime number $p$ divides the order of a group $G$, then there is an element $g$ in $G$ whose order is $p$. In addition, if $|G|$ admits a prime decomposition with a factor $p^k$, then Sylow's theorem tells us that a subgroup of order $p^k$ must exist.   My question is the following. If $|G|$ admits a prime decomposition with a factor $p^k$, does this imply that an element of order $p^k$ must exist in $G$?   Update: As explained in Mindlack's answer (among others), the implication does not hold and simple counterexamples exist. In the comments, Mindlack discusses special cases where such an element exists. JAXB Java generating XML, Why lowercase? Mass retagger badge I've been busy retaggin questions since I got my retag privileges.  Shouldn't there be an incentive to retag wrongly tagged questions, other than the urge to do things right?  I'm mean, the only "badge incentive" for retagging are "Firs edit" (Organizer) and "Creation of a tag used 50 times" (Taxonomist). The first one is easy to get, and the second one, right now in SO is almost impossible to get without cheating, that's editing 50 questions with an unlikely tag. Taxonomist I guess, could encourage wrongly tagging.  Is this really a problem?  We could have an "Tagged 200 question" badge, so people go for that badge, instead of taxonomist. If you get stuck in research, at what point in time should you ask for help? It's always worthwhile to take some time to try to figure it out by yourself. But eventually there's a point where you're unsure how much more time it would take to figure it out by yourself (which could take far longer than you originally think), and when asking for help could be more helpful.  So for analytical or computational research, what are some guidelines for when you should ask for help (from either your adviser, other grad students, or other researchers) when you get stuck? How do I enter a strong (long) DKIM key into DNS? I have a simple question.     over centuries / for centuries   what is the difference?  Thank you. I often use Cycles' built in Glass BSDF shader to achieve realistic refraction effects, but in many cases, it seems that Cycles' glass caustics creates an unfortunately large number of fireflies (pixels that are abnormally bright relative to their surroundings). In some models of mine, even when I render for 3000 or more samples/pixel, my final image is still covered in fireflies. While I can improve this in many cases by increasing the total amount of light available in the scene, this loses a lot of the potency of the refractive effects I'm looking for, especially in scenes with low light levels.  So my question is, how can I reduce the amount of fireflies while using Cycles' caustics without sacrificing the quality of the refractive effects?  Note: I am aware that Cycles' caustics will perform much better once Cycles has full support for bidirectional path-tracing, but it looks unlikely that that will become , unfortunately.