relationship between monomorphism and automorphism
An example for a homomorphism that is not an automorphism
Comparing $\pi^e$ and $e^\pi$ without calculating them Terminal, how to quit --More-- list Do professors care about their rating on websites such as RateMyProfessor or Koofers? I am wondering if a professor is already an Associate Professor, do they still care about their reputation in the eyes of the department considering tenureship is already granted? How to install Anaconda on Ubuntu? How to install  on Ubuntu?   Is there a way to use apt-get install?   I only have command line access to my server. How do I install Anaconda on Ubuntu 14.04 from the command line? Is it possible to make LaTeX mark overfull boxes in the output? Is it possible to make LaTeX mark overfull boxes in the output, for example by drawing red rectangles in the PDF where the boxes overflow? The purpose -- to more easily "see" where the problems are in the output. How to find all files with the same extension within a directory I want to search for all .conf files in /etc/. I tried grep -r *.conf /etc, but the result isn't right. What am I doing wrong? How to solve this recurrence $T(n) = 2T(n/2) + n\log n$ How can I solve the recurrence relation $T(n) = 2T(n/2) + n\log n$? It almost matches the Master Theorem except for the $n\log n$ part. Converting JSON data to Java object Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$. Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.   I was thinking of showing $A$ is not invertible, meaning it has an eigenvalue of $0$. Since no matter the power you give to $A$, it is still not the identity, you can deduce that at least one eigenvalue is indeed $0$. However this doesn't work if you have different eigenvalues in the Matrix, so I got stuck.  Help will be most appreciated. is there a way to request a new tag be created? Possible Duplicate:          I do not have sufficient reputation on any site to create new tags, but I do on occasion find myself in need of a tag that does not exists.  Is there a feature that allows for a person with sub-1500 reputation to request a new tag? When rolling percentages, do 1d100 and two d10s (percentiles) share the same probabilities? I know d100s are shunned because they take too much time to roll, and 1d10+1percentile is really fun, but do they share the same probabilities? Is it better to use a (digital) d100?  Also, do d100s have 0? You can't roll a 0 with percentiles, I think. $\bigcup \emptyset$ is defined but $\bigcap \emptyset$ is not. Why? Why does $\bigcup \emptyset = \emptyset$ but $\bigcap \emptyset$ is not defined?    If I had to guess, I'd say it's also equal to $\emptyset$. Difference between "Should" vs "ought to" It's essential that the documents (should)/(ought to) be destroyed immediately.   Which one is more suitable one as I know "ought to" and "should" are synonyms. Interaction of Invoke Duplicity and Misty Step The title of  of  is, "Is it posible [sic]?"  This doesn't provide any information at all. It should be mandatory for everyone asking questions on the Internet to read , which says:     Help your community learn from the question. When you field a good question, ask yourself "How would the relevant documentation or FAQ have to change so that nobody has to answer this again?" I found Sean Carroll's "A No Nonsense Introduction to General Relativity" (about page .  pdf ), a 24-page overview of the topic, very helpful for beginning study.  It all got me over the hump of learning the meaning of various terms associated with GR, most of which I had heard before without understanding.  It also outlined the most important examples.  Is there a similar document for quantum field theory, which presents the main equations, briefly describes the main ideas, and summarizes the most important applications and results so that the reader can feel the lay of the land before studying in depth? Do different "Picture styles" affect RAW output? I was  the manual of my camera (Canon 50D, but I am sure that each DSLR offer the same kind of functionality, albeit I don't know the exact names adopted) and noticed the issue of Picture styles which I had overlooked in my first reading.   In short I think that they are a collection of various calibration settings (more or less enhanced colours, things like that...). Those that are available on my camera are:   Standard Portrait Landscape Neutral Faithful  Monochrome 3 user definable modes.   These styles are so important to deserve a direct button just under the LCD screen of the camera (I guess this is useful for simplifying the use of the camera in the basic modes as opposed to manual / less automated ones).  And now for the question:   These settings obviously affect the appearance of the picture "as it comes out of the camera". But given the raw file I assume that they are all equivalent, a style is not applying a destructive transformation to the raw data, isn't it?  Shooting raw, is there a reason to favor one of these styles (Neutral?) over the other? Up to now I have been using Standard.   As a bonus (don't know if it deserves a separate question, in which case I will ask it again), what is the difference between Neutral and Faithful? $a\mid b,\ c\mid d\,\Rightarrow\ ac\mid bd $ $\ \, \bf\small [Divisibility\ Product\ Rule]$ I just need to check the reasoning in my proof is correct, I think it is valid although I'm not totally convinced because I can't follow the logic; does proving that $x$ is an integer prove that $ac|bd$?  Theorem: Let $a$, $b$, $c$, $d$ be integers. If $a|b$ and $c|d$, then $ac|bd$.  Proof: $aj=b$ and $ck=d$ for integers $j$, $k$. Then, $ac|bd$ implies $acx=ajck$ and thus $x=jk$ for some $x$. Since the product $jk$ is an integer, $x$ is an integer and thus $acx=ajck$ and thus $ac|bd$. One problem I have found when learning calculus is that there are many different ways to denote the derivative. If $y=f(x)=x^2$, then we could write \begin{align} f'(x)&amp;=2x \\ y'&amp;=2x \\ \frac{df}{dx}(x)&amp;=2x \\ \frac{df(x)}{dx}&amp;=2x \\ \frac{d}{dx}f(x)&amp;=2x \\ \frac{dy}{dx}&amp;=2x \end{align} And this is just Lagrange and Leibniz's notations alone. What I find troubling is that they all seem to be suggesting subtly different things about what the derivative actually is. Is it a function, a limit of a quotient, or both? In the interests of keeping my post brief, I'll focus my attention on $f'(x)=2x$ and $\frac{dy}{dx}=2x$, as these seem to be the most common notations.  $$ f'(x)=2x $$  It does make sense to think of the derivative as the gradient function: $$ f'\colon x\mapsto\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} $$ In this case the limit expression is equal to $2x$, and so we can write $$ f' \colon x \mapsto 2x $$ However, this notation seems a little counter-intuitive when we consider what it means to differentiate a function with respect to a variable other than $x$. If I ask what is the derivative of $f(x)$ with respect to $\frac{x}{2}$, does this question make sense? Is it simply $f'(\frac{x}{2})$? Or do we have to express $x^2$ in terms of $\frac{x}{2}$? And how can we can express this derivative using Lagrange's notation?  $$ \frac{dy}{dx}=2x $$  There are many things which are nice about Leibniz's notation, including the fact that it is explicit which variable you are differentiating with respect to. However, in this case, it is unclear whether we are talking about a function, or something else entirely. There are other issues. Some people say they dislike the Leibniz formulation of the chain rule $$ \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} $$ saying that they find it to be inaccurate. I don't really understand why this is the case. Could someone please elaborate? Prove that: $\sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{n}}}}<3,\,\forall n\in\mathbb N.$