I know that the one of the axioms of abelian categories is that the induced morphism $ \text{coker}(\ker f )  \longrightarrow   \ker ( \text{coker} f )  $ for any morphism $ f $ is an isomorphism. Let's call this axiom $ \textit{iso} $.  However, Vakil in his notes (, page 48) mentions axioms (2) and (3). One of my friends remarked that  $$(2) + (3)  \Longleftrightarrow  \textit{iso}  . $$   I have been able to show $ \textit{iso}  \implies (2) + (3)  $ , but how do I show that $  (2) + (3)  \implies  \textit{iso}  $?
Let's fix some terminology first. A category $\mathcal{C}$ is preabelian if:  1) $Hom_{\mathcal{C}}(A,B)$ is an abelian group for every $A,B$ such that composition is biadditive,  2) $\mathcal{C}$ has a zero object,  3) $\mathcal{C}$ has binary products,  4) $\mathcal{C}$ has kernels and cokernels.  A category $\mathcal{C}$ is abelian if it is preabelian and satisfies:  5) every monomorphism is a kernel and every epimorphism is a cokernel.  Define the coimage of a map to be the cokernel of its kernel, and the image to be the kernel of its cokernel. We have the following commutative diagram:    where $\overline{f}$ is the only existing map (because of universality of kernel and cokernel).  I'm having trouble proving the following:     A preabelian category $\mathcal{C}$ is abelian iff $\overline{f}$ is an isomorphism.   The converse is easily shown, I'm having trouble proving $\Rightarrow$...
Using vim with latex suite on two different computers, on one it's fine, but on the other my tikz pictures ''colapses'' (all text is put to the center of the figure, nothing is drawn) The same pc can compile fine from texstudio, but not from vim- latex suite  Edit: Seems the problem was compiling to dvi instead of pdf When estimating variance, why do unbiased estimators divide by n-1 yet maximum likelihood estimates divide by n? I am totally confused: On the one hand you can read all kinds of explanations why you have to divide by n-1 to get an unbiased estimator for the (unknown) population variance (degrees of freedom, not defined for sample size 1 etc.) - see e.g.  or .  On the other hand when it comes to variance estimation of a supposed normal distribution all of this doesn't seem to be true anymore. There it is said that the maximum likelihood estimator for variance includes only a division by n - see e.g. .  Now, can anyone please enlighten me why it is true here but not there? I mean normality is what most models boil down to (not least due to the ). So is the choice "division by n" yet the best choice for finding the best estimation for the true population variance after all?!? Is it true that $aH = bH$ iff $ab^{-1} \in H$ Let $H$ be a subgroup of a group $G$. The I know that for $a,b\in G$ we have $aH = bH$ if and only if $a^{-1}b \in H$.  My question is if it is also true that $aH = bH$ if and only if $ab^{-1} \in H$? Does it matter where the $-1$ goes?  I am guessing that this is not true since we keep putting the $-1$ on the left element, but I am also thinking that it might be true because they look like the same.  If $G$ is abelian, then it doesn't matter, so a possible counter example would have to involve a non-abelian group. If $H$ is a normal subgroup, then it should also be true because then $a^{-1}bH = Ha^{-1}b$. Google Chrome: Version 32.0.1700.107 m  This happens when displaying the profile and clicking on the number of visited days/consecutive.  . Checking if array is multidimensional or not? What is the most efficient way to check if an array is a flat array of primitive values or if it is a multidimensional array? Is there any way to do this without actually looping through an array and running is_array() on each of its elements? Momentum is defined by the product of mass and velocity. Now a projectile out of a gun has to have high velocity to penetrate a human body, as its mass isn't significant. But to reach this velocity, due to inertia/the law of energy conservation, momentum on both sides is to be equal. As there is strong negative acceleration in the opposite direction of the bullet, it should result in a strong force in the opposite direction of the bullet ($F = m*a $).  Therefore I wonder why shooting a bullet with a handgun is not ripping your hand apart. Can I visit the UK with an EU permanent residence card? I am non EU national, living in a Schengen country for past 5 years. I hold Indian passport and EU permanent residence card valid for 5 years.   Can i go to UK for few days to visit friends without a visa?  I am aware that UK is not in Schengen, and Schengen visas and residence permits issued by Schengen country is not valid for United Kingdom? But what about the PR card issued by a Schengen country?  I am also aware that EU citizens carrying EU passport have no restrictions to visit UK. What's needed for a complete backup system? For my new server, I want to setup a proper backup solution. I've found a great setup that will do twice-daily incremental backups via Dropbox. I plan on backing up my various databases, the webroot directory, the /etc directory/repository, and /var/log.   What else do I need to know to do a proper backup, and what is the standard setup here to ensure you can quickly restore from a backup in the case of a system failure?  I'm thinking of using Puppet, as it describes how the system should be. My restore procedure would look like this:   Install Puppet Run my puppet config Restore my backups from Dropbox (Should I create a script to do this? Probably)   This should also let me create a clone of my production server for use in dev environments, correct? Am I missing anything of importance? Add chat rooms for Android app Being able to join chat rooms from the Android app would be a plus! It would truly make us connected on the go, instead of having to sit down at our desks and open a browser window. This will allow more users to fully utilize the chat system and join events no matter where they are. Should a retrieval method return 'null' or throw an exception when it can't produce the return value? How do I delete my entire Stack Exchange account? Stack Exchange seems to have made it impossible to delete accounts. I can delete the accounts from other individual sites like Ask Ubuntu, but I can't delete the main account on stackexchange.com. I want to be rid of Stack Exchange completely! I don't want a Stack Exchange account whatsoever! I really don't like Stack Exchange and I want nothing to do with it. Please help me delete my stackexchange.com account! How to check if list item already exist when adding it with how to check Title( Country) if exists in the list. if exists it is necessary to do the update if not it is necessary to add it ?  function AddListItem() {      var listTitle = "Test";     //Get the current client context       context = SP.ClientContext.get_current();     var airportList = context.get_web().get_lists().getByTitle(listTitle);          //Create a new record       var listItemCreationInformation = new SP.ListItemCreationInformation();     var listItem = airportList.addItem(listItemCreationInformation);      //Set the values      var country= $("#Country").val();     var client= $("#Client").val();     // var nb1 = $("#nb1").val();      // var nb2 = $("#nb2").val();       // var nb3 = $("#nb3").val();      // var nb4 = $("#nb4").val();       // // var nb5 = $("#nb5").val();        listItem.set_item('Country', Country);     listItem.set_item('Client', Client);     listItem.update();     context.load(listItem);     context.executeQueryAsync(AddListItemSucceeded, AddListItemFailed); } function AddListItemSucceeded() { } function AddListItemFailed(sender, args) {     alert('Request failed. ' + args.get_message() + '\n' + args.get_stackTrace()); } Control Android with broken screen from PC Are these plural or singular? Given matrices $A$ and $B$, where that $AB = A + B$, prove $AB = BA$. I keep coming up with AB = AB. It seems like basic algebra, but for the life of me, I'm getting nowhere :/. Someone help please? Can someone explain the layer system? I am really confused with the different uses of the terms "Layers" and "Passes" in Blender. Can someone please explain how they work in Blender. I know about Render Passes like AO, Diffuse and so on, but there seems to be a confusion with layers. Also, how do the layer buttons work in the properties panel like in the image shown? I get it that you can assign objects onto certain layers, but whats the difference between "Scene" and "Layer" buttons in the image? How do they refer to rendering? How can you see what layer a certain object is on? Why do I not see evidence of discrete energy in the spectrum? The electromagnet spectrum looks to be continuous, yet energy is discrete. Shouldn't one see evidence of the electron's "quantum leap"? What's going on?     Moderator, please size my photo. Thank you. prints out a lot of very detailed protocol statistics like number of TCP reset messages received or number of ICMP "echo request" messages sent or number of packets dropped because of a missing route.  When in Linux netstat is considered deprecated at nowadays, then is there an alternative?  Statistics provided by ss -s are superficial compared to the ones provided by netstat. Find the last two digits of $3^{45}$ I was wondering if there is a simpler way to find the last to digits of a power such as $3^{45}$. I reduced it modulo 100 to get the answer, which is 43. But I was curious if there was a simpler, or more eloquent way to get this result.  Here's my method:  Using the fact that $3^4=81\equiv -19 \pmod{100}$ and $19^5=2,476,099\equiv -1 \pmod{100}$  $$ 3^{45}\equiv 3(3^4)^{11}\equiv 3(-19)^{11}\equiv 3(-19)(19^5)^2\equiv 3(-19)(-1)^2 \equiv -57 \equiv 43 \pmod {100} $$  I just felt like this was sort of a messy way to find out that the last two digits are 43. How can I install Sun/Oracle's proprietary Java JDK 6/7/8 or JRE? I want to install Oracle's JRE and to update to the latest version with the Software Updater when they released. Is there a Ubuntu package that is provided by Canonical or Oracle?  Before release Java 7, I followed  to install Java 6.  But it doesn't work for Java 7. There is no package sun-java7-xxx. How can you install Java 7?