Monic and epic implies isomorphism in an abelian category? Is it true that monic and epic implies isomorphism in an abelian category?
Some questions on abelian category Let  $f: C \longrightarrow D$ be a morphism in an abelian category $\mathfrak{A}$ with kernel and cokernel both zero. How can I show that it is an isomorphism? I am not able to find it's inverse.   Let $g: C \longrightarrow D$ be a morphism in the abelian category $\mathfrak{A}$. Let $i: K \hookrightarrow C$ be the kernel of $g$. How can I prove that the induced map $\text{coker}\ i \rightarrow D$ is a monic?   I don't want to use Mitchell's embedding theorem.
Explaining to laypeople why bootstrapping works Webform redirect I'd like to do something pretty straightforward: After user fills out a , I'd like a button that simply redirects them to a URL (in this case it's to a demo sandbox). Of course I'd want the user to fill in their info to capture for lead-generation purposes. My  is set up correctly in that it correctly captures all the user fields and submits them to me via email. However, upon completion "click" I'd like to have the user redirected to a determined internal URL www.mysite.com/foo  I can't figure out how this is done with the vanilla webform settings. Am I missing something obvious? Is there any way to have BibTeX or biblatex insert a bibliography without having citations in the body of the document itself? Maybe a way to hide the in-text citations, so that BibTeX still sees them and includes them in the bibliography, but does not print them in the body? How can I solve this functional equation, where $x,y$ are any real numbers and $f:\mathbb{R}\to \mathbb R$ is a function such that : $$f(x+y)+f(x-y)=2f(x)\cos y$$    I tried substituting $x=0$ to get $f(y)+f(-y)=2f(0)\cos y$ . Taking $x=y$ gives $f(2x)+f(0)=2f(x)\cos x$. I similar get some more relations like that, but its not really helping me in finding anything useful. I also think the function must either be $\sin$ or $\cos$ looking at the product to sum formulae. Can anyone tell me how should I solve this? For which $a \in \mathbb{C} -\ \{0,1\}$ does the equation $f(z) = f(az) $ has a non constant solution $f$ with $f$ being analytical in a neighborhood of $0$.  My attempt:  First, we can see that any such solution must satisfy:  $f(z)=f(a^kz)$ for all $k \in \mathbb{N} $.  If $|a|&lt;1$:  The series $z_{k} = a^k$ converges to 0 which is an accumulation point, and $f(z_i)=f(z_j)$ for all $i, j\in \mathbb{N} $. Thus $f$ must be constant.  If $|a|=1$:  For all $a \neq 1$ , $f$ must be constant on any circle around $0$, so again $f$ must be constant.  My quesions are:   Am I correct with my consclusions?   Also, I'm stuck in the case where $|a|&gt;1$. Any ideas?  Thanks I've deleted a system file from my Mac and need to get it back.  System files can include:   hidden files in the root of / - especially unix files in /private  /System files /Library files /Applications files that ship with the core OS   System files are not any app, user files or files that can be regenerated by a known sequence such as cache files, Spotlight index files, etc…    Canonical question and answer for . std::auto_ptr to std::unique_ptr Yesterday I upgraded my Ubuntu 18.10 to Ubuntu 19.04, and after the restart, the trash icon and my Desktop files have disappeared from the UI. If I go to the Desktop folder from my personal folder, I can see the files, but my Desktop is empty.  How can I show my files in the Desktop again? "Dummy Output" No sound in Ubuntu 16.04 None of the HDMI output are working, all saying "dummy output", it was working yesterday, but when I use the headphones it became soundless, with or without the headphones.  Thank you already! How to execute code after html form reset with jquery? I have heard from people who have switched either way and who swear by the one or the other.  Being a huge Eclipse fan but having not had the time to try out IntelliJ, I am interested in hearing from IntelliJ users who are "ex-Eclipsians" some specific things that you can do with IntelliJ that you can not do with Eclipse.  Note: This is not a subjective question nor at all meant to turn into an IDE holy war. Please downvote any flamebait answers. table of contents no word separation Is it possible to have in the toc no word separation but have a word separation in the chaptername:  For example:   Contents  1. max Mustermann   Chapter  Max Muster- mann I gave a scammer my telephone number, address, and name. How bad is it? Updated question because old one was embarassing: I saw a spam email from "PayPal" and I didn't realize that it was a phishing attempt. I gave the phishing site my name, telephone number, and email. What could he do with them?  Old question: Well, I saw a spam email, but I didn't realize that it's really a scam at first. I gave him my telephone number, address, and name. What would he do with them? Could he ruin ALL of my reputation with it? Mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$? What is the mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$?  It is interesting to note that Gaussian R.V here is zero-mean and non-central Chi-square Distribution doesn't work.  Thanks. When should one use the "recent past"? How much time should have passed? match first column of file a with paragraphs of file b I have 2 files. The first, fileA looks like  TCONS_00000066  XLOC_000030     -       u       q1:XLOC_000030|TCONS_00000066|0|0.000000|0.000000|0.000000|0.000000|- TCONS_00000130  XLOC_000057     -       u       q1:XLOC_000057|TCONS_00000130|0|0.000000|0.000000|0.000000|0.000000|- TCONS_00000395  XLOC_000206     -       u       q1:XLOC_000204|TCONS_00000393|0|0.000000|0.000000|0.000000|0.000000|-   FileB looks like:  &gt;TCONS_00000001 gene=XLOC_000001 AGATGAGCTGGTGGGGATGCTCTAAGAGAACGAGAGAAGCACAGAGCAGATAAACCACACCCACAGGCAC CACCGTCCTTGTTGGTAATGAAGAAGACGAGACGACGACTTCCCCACTAGGAAACACGACGGAGGCGGAG ATGATCGACGGCGGAGAGAGCTACAGAAACATCGATGCCTCCTGTCCAATCCCCCCATCCCATTCGGTAG TTGGATTGAAGACTACCGAATAAGAGAAGCAGGCAGGCAGACAAACCCTTGAACCAAGGAGTCCTCGCTG AGGAAGCTTTGGATCCACGACGCAGCTATGGCCTCCCCGCCCACCAGGCCGCCAGCCACAACCAGCTGAC TAGGTCGCATGCATCATCAGATTTCAATCTCCCTTCGTTCCCTGTCCCTAATCCAATACCAATAGGGAGC AATCAGCTGCTCCTCGACGGCGAGGGAGATGTCGTCGGCCGCGGGCCAAGACAACGGAGATACCGCTGGG GACTACATCAAGTGGATGTGCGGCGCCGGTGGCCGTGCGGGCGGCGCCATGGCCAACCTCCAGCGCGGCG TTGGCTCCCTCGTCCGTGACATTGGCGACCCCTGCCTCAACCCATCCCCCGTTAAGGGGAGCAAAATGCT CAAACCGGAAAAATGGCACACATGTTTTGATAATGATGGAAAGGTCATAGGTTTCCGTAAAGCCCTAAAA TTCATTGTCTTAGGGGGTGTGGATCCCACTATTCGAGCTGAAGTTTGGGAATTTCTTCTTGGCTGCTATG CCTTGAGTAGTACCTCAGAGTATAGGAGGAAACTAAGAGCTGTTAGAAGGGAAAAATATCAAATTTTAGT TAGACAGTGCCAGAGCATGCACCCAAGCATTGGTACAGGTGAGCTTGCTTACGCTGTTGGATCAAAGCTA   Now, fileA contains selected transcript numbers in the first column and fileB contains sequences of all transcripts. I want to scan fileB for the first column of fileA and print the trailing sequences of matching transcripts along with the transcript number. In an older question from June 2011, , apparently revived by the system, a user is asking about the origin of the rotation of celestial bodies.  The  (with 17 votes) states the following:     When things collapse under their own gravity in space (i.e. clouds of gas and dust), any small amount of asymmetry in the collapse will be enough start it spinning.   And then it goes on saying:     Even if it spins by a tiny amount, as it collapses, angular momentum     conservation will mean it spins more and more quickly.   Is all this really correct?  I could try to ask in a comment to the answer. I actually did. But the author is no longer registered.  Hence I am asking it as a question.  But that is not the end of the story. As I was about to ask my question, I looked for other relevant questions, and there are some. I did not make an exaustive survey, but I did not find one that had satisfactory answers:    (december 2011) has just one accepted answer (0 votes) stating essentially that the solar system has angular momentum because it started from a cloud that had angular momentum. No comment.  (july 2011) is questionning about everything from galaxies to electrons. The answers are of a much higher quality, but while they seem to justify the possiblility of spin and angular momentum (and its conservation), there are not clear on how it appears, at least for celestial bodies.  (june 2013): That was my own very naive first question on this site. It is related but does not directly address spin.   One answer I found is that momentum appear from bodies in relative motion past each other. Nice, but not really an answer: where did they get their relative speed to begin with. Unfortunately the answer is: from the angular momentum of larger structures they belong to (well, it can be more comples). So this is just chicken and egg explanation. But how did it get started?  Note that my question is not a duplicate, since it is primarily whether the above statements are correct, which is not considered in these other questions.  Furthermore, these questions are fairly old, and given that none had received answers that seem to answer the question of the origine of existence of celestial angular momentum (not in a way I can understand anyway), it may be appropriate to attempt restarting the process, with the above discussion to motivate that an answer should explain the physical mechanism that started the existence of spin of celestial bodies. Proving a function is Lipschitz continuous Show that the following function is Lipschitz continuous and find a Lipschitz constant  $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{  , } |x-1|\leq\frac{1}{2}\text{  , } |y-e|\leq\frac{e}{2} $$  I have no clue on where to begin to prove this.   My first question is how I should interpret this function, is $x$ a sort of constant? Is there a bijection of the natural numbers which swaps $\frac{1}{n}$-summable subsets with $\frac{1}{\sqrt{n}}$-summable subsets? Let me start with a precise statement of the question.  For a subset $A\subseteq \mathbb{N}$ and a series of positive real numbers $\sum_{n=0}^\infty a_n$, I'll use the notation $\sum_A a_n$ as a shorthand for $\sum_{n\in A} a_n$.  Is there a bijection $f:\mathbb{N}\rightarrow \mathbb{N}$ with the property that for every $A\subseteq \mathbb{N}$, $\sum_A \frac{1}{n}$ converges iff $\sum_{f(A)} \frac{1}{\sqrt{n}}$ converges?  Background: Fix a series of positive terms $\sum_{n=0}^\infty a_n$.  Given a subset $A\subseteq \mathbb{N}$, call it $a_n$-small if $\sum_A a_n$ converges.  The following proposition is easy to prove:  Proposition:  The set $\{A\subseteq \mathbb{N}: A\text{ is } a_n\text{-small}\}\cup \{\mathbb{N}\}$ is a topology of closed sets on $\mathbb{N}$.  I will use the notation $(\mathbb{N},a_n)$ to refer to this topology.  Then one can ask how topological properties of $(\mathbb{N},a_n)$ are related to series properties of $\sum a_n$.  For example, I can show:  Proposition:  The following are equivalent.  $\sum a_n$ converges.  $(\mathbb{N},a_n)$ is discrete.  $(\mathbb{N},a_n)$ is disconnected.  $(\mathbb{N},a_n)$ is Hausdorff    And there are other nice things.  For example, $(\mathbb{N},a_n)$ is compact iff $(\mathbb{N},a_n)$ is cofinite iff $\liminf a_n &gt; 0$. With this language, my question can be reformulated as...  Are $\left(\mathbb{N}, \frac{1}{n}\right)$ and $\left(\mathbb{N}, \frac{1}{\sqrt{n}}\right)$ homeomorphic?  I have made very little progress on this.  Of course, the identity function $i:(\mathbb{N}, \frac{1}{\sqrt{n}})\rightarrow (\mathbb{N}, \frac{1}{n})$ is a continuous bijection, but the inverse map is not continuous.  Also, if there is such a bijection, there there is such a bijection with agrees with $i$ on any preassigned finite set. Edit  I thought I add in a slightly suprising (to me, at least) example when things work out to be homeomorphic. Begin with a convergent series $\sum c_n$. and divergent series $\sum a_n$ with $\lim a_n = 0$.  Create a new series $b_n$ using all the terms of $c_n$ and $a_n$ (in whatever order you wish).  Then $(\mathbb{N}, a_n)$ and $(\mathbb{N}, b_n)$ are homeomorphic.  The idea is that since $\lim a_n = 0$, there is a convergent infinite subseries $\sum_A a_n$.  Then we use a bijection which with $A$ and $A\cup \{\text{indices of }c_n\}$ to &quot;squeeze&quot; the $c_n$ in without changing the topology.  Of course, I am glossing over many details, but I can include them if desired. Can a PC with the Alert feat alert his friends? Assume the following situation: The party has been surprised by an ambush. Every PC is surprised except PC Bob because he has the Alert feat. Thanks to his high initiative bonus Bob gets to go first. He shouts "look out" to alert his party members of the imminent threat. The idea is that his party members are no longer surprised because he warned them.   While I think that this makes sense from a storytelling perspective, I am afraid this would be too strong. as it allows one PC with the Alert feat to give the "not surprised" benefit to other members of his party pretty much for free.  Is this allowed? Are there rules or an official ruling or similar on this?  If not, do you have any experience with this situation, and how did it play out in your games?