I understand that the zero ring can't be a zero object for the category of rings, because in that case the 'arrows' are ring homomorphisms which, by definition, but maintain the unit.  But in the category of sets, the 'arrows' are just functions between sets. So why isn't the empty set a terminal object (and hence a zero object)?
From Awodey: In any category $\mathsf{C}$,  an object $0$ is initial if for any object $C$ there is a unique morphism $0 \to C$,  an object $1$ is terminal if for any object $C$ there is a unique morphism $C \to 1$.  First I was puzzled by the fact that we can do mappings from a zero set to any other set in a category $\mathsf{Sets}$. But then after I accepted it conceptually, I do not understand why there is no unique morphism to an empty set from every object in a category $\mathsf{Sets}$. Is this because we cannot map a non-empty set to an empty set (since our mapping is total) although we can map an empty set to a non-empty set?  Please, note, I am familiar with the proposition saying: Initial (terminal) objects are unique up to isomorphism.
Why does the introduced part-title in letter appear twice? When can lands be tapped? Downloading files using wget I am trying to download files from .  The URL is:   When I use this command:  wget http://www.ncbi.nlm.nih.gov/geo/download/?acc=GSE48191&amp;format=file    I get only index.html?acc=GSE48191 which is some kind of binary format.  How can I download the files from this HTTP site? proof that $f'$ is continuous at $a$ Let $f:I \rightarrow \mathbb{R}$ derivable in set I. if $\lim x_n = \lim y_n =a \in I$ with $x_n \neq y_n$ and $\lim \frac{f(y_n)-f(x_n)}{y_n - x_n} = f'(a)$ proof that $f'$ is continuous at $a$  my attempt $x_n \neq y_n$ so by MVT $\exists z_n \in I$ like $x_n \geq z_n \geq y_n$ with $\frac{f(y_n)-f(x_n)}{y_n - x_n}=f'(z_n)$ for each $x_n$ and  $y_n$ then $f'(a) = \lim\frac{f(y_n)-f(x_n)}{y_n - x_n}= \lim f'(z_n)$   Using $\lim x_n = \lim y_n = a$ and $x_n \geq z_n \geq y_n \Rightarrow z_n \rightarrow a$  if $ \lim_{x\rightarrow a} f'(x) \neq f'(a)$ then $\exists \epsilon &gt; 0$ such $\forall n\in \mathbb{N}$, $z_n \in [x_n,y_n]$ $0 &lt; |z_n - a| &lt; \frac{1}{n}$ and $|f'(z_n) - f'(a)| \geq \epsilon$. So $z_n \rightarrow a$ but will do not have $\lim f'(z_n) = f'(a)$  so $\lim f'(z_n) = f'(a) = \lim_{x\rightarrow a} f'(x)$ when $z_n \rightarrow a$  and thats implies $f'$ continuous at $a$  my question is, that's correct ? if not, can you help me? thanks Include the option to flag posts as spam in the Low Quality queue I guess this is an additional to previous requests like  and .  But I just came across  review item, and was unable to flag it as spam.  edit: Since I posted this, the item has been removed (deleted) here is a screenshot: launch a VM on Ubuntu On Ubuntu 14.04, I installed VirtualBox on which I have created 2 Ubuntu virtual machines.  Does Ubuntu provide a way to launch  the virtual machine I want at a particual moment ? Is there any solution for this problem by developing a .sh script for example or using any other programming language ? Is there a way in javascript to detect if the unload event is caused via a refresh, the back button, or closing the browser? I am currently looking at the "unload" event of a window to try to determine how the "unload" event was triggered, but am having little success. Is there a way to determine how the javascript event was triggered?   Page Refresh Back Button (or navigate away from the page) Closing the Browser   Essentially I need to execute some code only when the browser window is being closed, not refreshed or navigated away from.  Purpose: When a customer does an update of our software, the update will redirect their first Internet request to an offer page. There is a button for a "Do Not Bother" option, but some users will simply close their browser. Upon closing the browser, I need to duplicate the "Do Not Bother" functionality so the user no longer gets redirected to the offer page. Simply attaching to the "unload" event will not work due to the different ways of leaving a page. php--glob for searching directories and .jpg only How do you handle aspect ratio differences with Unity 2D? I've gotten a lot of answers to this question, but they are all generic and generally not very useful.  None of the tutorials talk about aspect ratio and dealing with mobile devices and there are a zillion ways to do it, all seem to have gotcha's and flaws.  I really would love to know what successful games have used to handle different aspect ratios on iOS and Android without making a zillion different sized assets.  I am strictly speaking mobile, not desktop, specifically with Unity and I don't care about the UI, I only care about the gameplay canvas.  Issues I have in mind is when there are key things that have to be in certain places and cannot fall off the screen.  Using black bars on top or bottom is unacceptable these days. Convergence of $\int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx$ Where can I find a Java to C# converter? I needed to convert a Java 1.5se app to C# 2.0.  Does anyone know of a tool (preferably free/open source) to do this? Nested roots sequence, how to prove it's monotone and bounded? Let $a\ge1$ and define the sequence $(x_n)$ recursively by: $$x_1 = \sqrt{a}$$ $$x_{n+1}= \sqrt{a+x_n}$$  Here's what I did:  Plugging in some values makes it seem as if the sequence is increasing. I first solve the equation $t=\sqrt{a+t}$, equivalently $t^2 -t -a =0$, to get an idea what's going on. The roots are $$t_{1,2}= \frac{1 \pm \sqrt{1+4a} }{2}.$$ Therefore, between the roots, that is for $$t\in \left( \frac{1 - \sqrt{1+4a} }{2}, \frac{1 + \sqrt{1+4a} }{2}\right)$$  we have $t^2-t-a \le 0 \iff t \le \sqrt{a+t}$. Here's my attempt at showing inductively that $$\forall n\in \mathbb{N}, \ x_n \le \frac{1 + \sqrt{1+4a} }{2}$$  For $n=1$, the statement is true (simple arithmetic). Suppose it's true for $n$. Consider  $$x_{n+1}= \sqrt{a+x_n}\le \sqrt{a+\frac{1 + \sqrt{1+4a} }{2}}=\frac{1 + \sqrt{1+4a} }{2},$$  because $\frac{1 + \sqrt{1+4a} }{2}$ satisfies $x=\sqrt{x+a}$.  So this establishes the sequence $(x_n)$ is bounded. Now, by induction again, I try to show it's monotonically increasing. The base case $x_1\le x_2$ is obvious. Assume (strong induction) $x_{n-1}\le x_n$ for all $n&gt;1$. But since we have $$x_1\le x_n \le \frac{1 + \sqrt{1+4a} }{2}, $$   then $x_n$ lies in the appropriate interval, so $x_n \le \sqrt{a+x_n}=x_{n+1}.$  Is this proof correct? Can the condition on $a$ be relaxed to $a&gt;0$? Maybe even small negative $a$? And could someone provide me with another proof of convergence of $x_n$? I absolutely hate this one! General trigger bulkification - best practices A general question asked from time to time on stack exchange is question of trigger bulkification.  Therefore I have 2 questions:   Which best practices are you using? What is the best way to handle updates of more than 200 records? (In case of more than 200 records the trigger is executed more than once.) Continuous function on closed unit ball Take a continuous mapping $f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where $\bar{B^{n}}$ is a closed unit ball in $\mathbb{R}^{n}$. Assume that $f(x) \neq x$ for every $x \in \bar{B^{n}}$. Define another function $r$ by following the directed line segment from $f(x)$ through $x$ to its intersection with $\partial B^{n}$, and let the intersection point be $r(x)$. Is it immediately evident that $r$ is a continuous function? Is so how does it follow? Thanks. How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero? This is part of a homework assignment for a real analysis course taught out of "Baby Rudin."  Just looking for a push in the right direction, not a full-blown solution.  We are to suppose that $f(x)f(y)=f(x+y)$ for all real x and y, and that f is continuous and not zero.  The first part of this question let me assume differentiability as well, and I was able to compose it with the natural log and take the derivative to prove that $f(x)=e^{cx}$ where c is a real constant.  I'm having a little more trouble only assuming continuity; I'm currently trying to prove that f is differentiable at zero, and hence all real numbers.  Is this an approach worth taking? How many Shutter Clicks on a New Camera considered normal? How many Shutter Clicks on a New Camera considered normal?    I purchased a quotes new Nikon D750 and checked the shutter count and it has 264 Shutter Clik count on it.    Is that suspect of a returned purchase or used camera? How to connect to my web server? mkdir -p for files Find the last $2$ digits of $$\large7^{7^{7^{10217}}}$$   So far I have: $17\cdot 601=10217$ and $7=7 \pmod{10}$. Any help greatly appreciated. How do I convert two lists into a dictionary? Imagine that you have the following list. keys = ['name', 'age', 'food'] values = ['Monty', 42, 'spam']  What is the simplest way to produce the following dictionary? a_dict = {'name': 'Monty', 'age': 42, 'food': 'spam'}