Why does abstract algebra have just binary functions?
Is abstract algebra (mostly?) restricted to $2$-ary operators?
How do I install VMware Player 4.0.3 (and patch the installer to work on my 12.04 system)? When I cd a link, my current path is prefixed with the link's path, rather than the path of the directory the link links to.  E.g.  ~/dirlinks/maths$ ls -l logic lrwxrwxrwx 1 tim tim 71 Jul 27 10:24 logic -&gt; /windows-d/academic discipline/study objects/areas/formal systems/logic  ~/dirlinks/maths$ cd logic ~/dirlinks/maths/logic$ pwd /home/tim/dirlinks/maths/logic  ~/dirlinks/maths/logic$ cd .. ~/dirlinks/maths$   I would like to have my current path changed to the path of the linked dir, so that I can work with the parent dirs of the linked dir as well. Besides ls the link to find out the linked dir, and then cd into it, what are some simpler ways to accomplish that? For example, after cd into a link, how do you change your current path to the path of the linked dir? Why do people buy stocks that pay no dividend? Installing MinionPro fonts on Ubuntu 10.04 running TexLive 2012 I followed the instructions to install MinionPro fonts from this website.   However, when I tried to run the tex file at the bottom, I have this error message:  ! LaTeX Error: File `MinionPro.sty' not found.   Running the command kpsewhich MinionPro.sty gives the following output:  /usr/local/share/texmf/tex/latex/MinionPro/MinionPro.sty   So the system knows where MinionPro.sty is, so why can't the file compile? Not every open set in $\mathbb{R}^n$, for $n\geq2$, is a disjoint union of countable open balls. I know that every open set in $\mathbb{R}$ can be decomposed as the union of countably many disjoint open intervals and I'm trying to show that this property no longer holds in higher dimensions. For this I'm trying to show that there is no countable, pairwise disjoint open balls whose union is the open square $(0, 1)^2 ⊂ \mathbb{R}^2$. I know that it is because of the corners, but I don't know how to write this in mathematical terms. Thanks for any help. Transparent background but solid title bar and text in Windows 7 I'd like to have a terminal like the one in Ubuntu where the background is transparent but title bar and text is still solid. I have tried Console, but setting the alpha value makes the entire window transparent, which makes the text hard to see.  Is that possible? Calculating geometry for feature class with x/y and Lat/Long values in different projections? I see this a lot, someone in a category of people attempts to shut down anyone with an opinion that is not from the same category. For example, people without children weighing in on parenting; the parent may say something like "well, since you don't have kids and I do, my position is of more validity than yours."  Another example: a drug user (of some kind) telling someone who has never done drugs and does not care to, "until you tried it, you have no right to judge me."  People frequently try to limit the consideration on opinions to those in their same group, while it is perfectly reasonable that the non-drug user, especially if they are a chemist or well read on the subject, may have an equally valid opinion, or even more valid. In fact some might even say that someone NOT in the category has a better chance of being objective and seeing the forest for the trees.  So what is the fallacy of trying to shut down opinions when the source of the opinion is not someone who has personally had the experience under discussion? I can't really remember how long ago it was, but I think the book was marketed towards children/teens.  All I remember is the basic premise of the story.  Some story about these unusual little blue electric critters that only certain people could see, and these kids took it upon themselves to go about destroying them since they fed off energy or something like that.  Not sure whether they fed off people or off electricity... or both maybe?  I think the dominant method of destroying them was an electric baton that was taken from law enforcement.  I wanna call them buzz batons, but that's probably thanks to Artemis Fowl. How can I start to learn web development? I'm very interested in learning web development and have learned some basics regarding HTML and CSS a few years back from W3Schools, but I never learned how to develop websites that are up to today's standards, i.e. free-flowing sites without the use of tables to govern the design of the site.  What are some good resources that I can start with? How do I edit an invalid sudoers file? It throws the below error and it's not allowing me to edit again to fix it.  Here is what happens:  $ sudo visudo &gt;&gt;&gt; /etc/sudoers: syntax error near line 28 &lt;&lt;&lt; sudo: parse error in /etc/sudoers near line 28 sudo: no valid sudoers sources found, quitting Is there a ZIP code I can enter when paying-at-the-pump in the USA with a foreign credit card? In the USA, many  fuel dispensers that accept credit cards require a  that corresponds to the billing address of the credit card.  In Canada, for instance, we have , so it's not obvious what you're supposed to type in since the gas dispensers have only numeric keypads.  Entering '00000' or '90210' seems to always result in failed authorization.  Is there a zip code that works for credit cards with foreign billing addresses? Understanding the differential $dx$ when doing $u$-substitution Meaning of "kernel" In analysis, there are at least three kinds of "" concepts:   In probability theory, there is a concept called , also called probability kernel, from one measure space $X$ to another $Y$. It is actually a family of measures on $Y$, indexed by members of $X$. It can transform a measure on $X$ to a measure on $Y$, in terms of integration.  A special case is  of probability density function in statistics, where the  is actually a probability kernel. In real/complex analysis, there is a concept called  from a product space $X \times Y$ to $\mathbb{R}$ or $\mathbb{C}$. It can transform some special function: $X \to \mathbb{R}$ or $\mathbb{C}$ to another one:$Y \to \mathbb{R}$ or $\mathbb{C}$. This is used for defining integral transformation In Hilbert space theory, there is concept called , which is a family of bounded mappings for a family of Hilbert spaces. For every two Hilbert spaces $X$ and $Y$ in the family, there are exactly two members of the kernel, one mapping from $X$ to $Y$ and the other from $Y$ to $X$.   I wonder    Are they related concepts, since they seem to share some indescribable similarity? Can they be unified?   In particular, the first two are similar to each other as the kernels induce transformations on measures and on functions in terms of integrals.  It is however not obvious to me how the third one, i.e. the Hilbert space one, is related to the second one. Are they related to the  concepts? Or even to the  concept?   I feel like some insights and references for catching the general ideas shared across as many kinds of "kernels" as possible.  Thanks and regards! php--glob for searching directories and .jpg only Induction on a sum The left hand side has terms involving  $\binom{n}{m}= \dfrac{n!}{(n-m)!m!}$    $$1+\dfrac{1}{2}\binom{n}{1} +\frac{1}{3}\binom{n}{2}+...........+\frac{1}{n+1}\binom{n}{n} = \dfrac{2^{n+1}-1}{n+1}$$  I've done induction and proved $P(0)$ holds and also $P(1),P(2)$ holds (Just in case)  Now I've found $P(K+1)$ so the sum on the left is going to equal $\dfrac{2^{n+1}-1}{n+1}$ + $\dfrac{1}{n+2}$  and the right hand side is going to equal $\dfrac{2^{n+2}-1}{n+2}$  My problem is coming with the algebra though. I can't get those sides to equal. I can't get rid of the $n+1$ in the denominator.   Anyone have any ideas for me? They'd be much appreciated!  P.S. Someone gave me a hint and said that you can use the binomial theorem to solve this. How to make a different font and color of text in labels? How do I make my rim brakes stop squealing? My rear rim brakes are squeaking loudly. I don't know what the problem is, but I am sure it has nothing to do with rain because it happens all the time even if it is dry.  Anybody have an idea what could be problem and how to solve it? It makes the rear brakes kind of useless, because if I use them I will scare the people or drivers around me. I'm confused about the use of symbols $\vdash$ and  $ \models$. Reading the answers to  and  I see that:  the turnstile symbol $ \vdash $ denotes syntactic implication. Then $ S \vdash \psi$ means that $\psi$ can be derived from the formulae in $S$  the double turnstile $ \models $, denotes semantic implication. Then the notation $A \models B$ means simply that B is true in every model in which A is true.  So I understand that, e.g., $(ZFC+ A) \vdash B$ means that $A \Rightarrow B$ is a theorem in ZFC and, in this sense $\vdash$ is a mathematical symbol for a theorem. While $(ZFC+ A) \models B$ means that $ B$ is true in every model of ZFC in which $A$ is true, and, in this sense $\models$ is not a mathematical but a metamathematical symbol.  I understand correctly? And those are the only uses of the two symbols? For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?