Find all solutions to $a^b = b^a$
$x^y = y^x$ for integers $x$ and $y$
ArcGIS Serious Application Error I'm a complete novice to GIS and just began an Introductory course. I am having an issue when I try to access the Symbology tab on a layer. Specifically, when I click on the 'Categories' field and 'Unique Values' subfield, ArcMap crashes and comes up with a report window to send to ESRI. The message is      ArcGIS for Desktop has encountered a serious application error and is   unable to continue.   What could be the problem? View referrals from one part of my site to another in Google Analytics I have an online shop with a blog structured this way:   Shop: example.com Blog: example.com/blog   Both index have the same Google Analytic Code (UA). In the blog I write articles about some products with a link to the product page.  Now the problem is, I want to see stat on visits whose referral links have this pattern example.com/blog/*. I am looking for a positive, passionate word to describe the all-round capabilities of a person.   Are there any words better than 'all-rounder'? Can anyone help me identify a story about kids with silver blood? Suppressing addition of blank page before figure when wrapping long JPEG image I believe in Windows Vista, we could use linkd command. However, I cannot find this command in Windows 7. I know I can use Windows Explorer and delete the junction, but I wonder if this can be done in Command Prompt.  I can use mklink command to create a junction from a Command Prompt, so, what is the opposite of it (something like linkd)? Antivirus software is breaking SE web page? (blocking CSS and JS)? Remove default ID attribute from HTML helpers in ASP.NET MVC Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain I want to show that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw)\subset\mathbb{A}^4$, is not a unique factorization domain.    Morally, all we need to do is find some nonzero element that can be written in two distinct ways as a product of irreducible elements, but nothing is coming to mind immediately. Thoughts? Real Analysis question on FTC, Integral Let $g:[0,1] \rightarrow \mathbb R$ be a continuous function and assume that    $$ \int_{0}^{1} g(x) \phi'(x) dx = 0 $$   for all continuously differentiable functions $\phi: [0,1] \rightarrow \mathbb R$, where $\phi (0) = \phi(1) = 0$. Show that $g(x) = c$ for all $x \in [0,1]$ for some constant $c \in \mathbb R$.    My thoughts are applying integration by parts with $G(t) = \int_{0}^{t} g(x) dx $, since now we get  $$ 0 = \int_{0}^{1} G(t) \phi'(t) dt  + \int_{0}^{1} g(t) \phi (t) dt $$  which doesn't really go far. The second thought was applying MVT for definite integrals, which doesn't go far either as the conditions are hardly satisfied.   Any hints? Conjectures Involving floor(x)/piecewise continuity with regards to Integration and Differential Equations [This answer has been heavily edited in response to a long chain of comments on Eric Stucky's answer.]  I came up with these few theorems and I am curious whether or not my hypothesis are true. I don't really have a method for proving these, mind you. I'm more considering these as things I've noticed to be true. I just don't know whether they are always true.  Some of these are definitions to which I will append "def." to the end of the title. Think of these as a given. It's like how the integral is defined abstractly as area. I'm defining terms.  Jump Series Theorem def.  A function $f$ [consisting of floor] is said to have a jump series if there is a series $\Sigma=\sum_{n=1}^\infty a_nH_n$ of scaled step functions such that $f+\Sigma$ no longer has any discontinuity attributable to floor. The sequence of scaling factors $a_n$ is called the jump sequence for the function.  Definition of Implied Derivative def.  We define the evaluation of the implied derivative of a function $y$ at the point $x$ as the set of values $y^{\to}(x) = \{y | y = \lim_{h \to 0^+} \frac {y(x+h) - f(h)}{h} \lor y = \lim_{h \to 0^-} \frac {y(x+h) - f(h)}{h}\}$. It is in this sense that the implied derivative has the potential of being a multi-valued operator.  Jump Series Resolution Theorem  [The original wording is preserved; I have a formalism but it assumes that having a jump function is equivalent to being piecewise continuous, which I'm not sure is true.]   The integral of a floor-based function is floor treated as a constant minus the appropriate jump series of the integral.  Integration Continuity Theorem  The (ordinary) indefinite integral of any function $f$ having a jump series must be continuous (or fixably discontinuous) on the domain of $f$. However, this can vary for the implied indefinite integral.  Calculus Jump Theorem def.  [Again, the original wording is preserved; the meaning of this phrase has not been properly illuminated]  Any form of jumping within a function representable as a floor function can be considered as a portion of the constant of the implied integration for floor.  Jump Location Theorem  For any function $a(x)$ with a jump series, the function $\lfloor a(x)\rfloor$, has points of discontinuity precisely at those $x$ such that $a(x) - [a(x)] = 0$.  Composite Floor Function Jump Sequence Theorem One  If $a$ is a function such that $a(x) - \lfloor a(x) \rfloor = 0$ when $x - \lfloor x \rfloor = 0$, then the jump sequence of the composite function $f(x - \lfloor a\rfloor)$ is a constant value $f(0) - f(1)$.  Implied Differential Equations Conjecture  Any continuous solution to an equation constructed with implied derivatives of various orders is the same as the solution to the similarly constructed differential equation. I want my prompt to display:   current (absolute) working directory, colored in green and in a NEW line, the dollar sign and an empty space.   I am using this line in .profile  export PS1='\e[0;32m$(pwd)\n\e[m$ '  So when in e.g. Desktop, my prompt looks like this:    However, sometimes when hitting the up arrow to re-run previous commands, at the start of the prompt a random char sequence appear that does not seem to go away unless I hit enter, e. g.    I have never hit a cd cd Desktop command. The weirdest part is that backspace won't even delete the first one of the two cd commands above!  Any suggestions? Why does Google not consider dot in usernames of Gmail addresses? E.g.: If your Gmail ID is abc.xyz@gmail.com, it considers this the same as abcxyz@gmail.com.  Why is this so? Pros and Cons of a Decentralized Puppet Architecture We have around 300 RHEL servers that are currently connecting to a Puppetmaster server.  However, we have noticed some performance bottlenecks and it is the point of failure in our system.  I am fairly new to puppet in general and I am considering creating a decentralized puppet architecture instead of having Puppet clients connect to the Puppetmaster server.  Aside from what I would suspect to happen such as performance gain and lack of signing and exchanging SSL certs for new machines, what are other pros and cons to setting up a decentralized architecture? TexMaker (and TexShop) not working after installation of El Capitan Let $T,S$ be linear transformations, $T:\mathbb R^4 \rightarrow \mathbb R^4$, such that $T^3+3T^2=4I, S=T^4+3T^3-4I$. Then S is:   one-one but not onto onto but not one one invertible non-invertible   (One or more correct options)  My attempt: $S=T^4+3T^3-4I=T(T^3+3T^2)-4I=T(4I)-4I=4T-4I$  How do I go about proving or disproving my options?   My thoughts on options # 1. and 2.: I'm guessing $S$ is one-one if $Ker(S)=\theta$ is one way to go, but how do I obtain the kernel when I don't know what the transformation is? $Ker(S)=\theta$ if $T=I$, which does satisfy $T^3+3T^2=4I$. But so can other $T$s.   My thoughts on options # 3. and 4.: $S$ is invertible if $det(S)\neq0$, which is possible $det(T-I)\neq0$, i.e. $T\neq I$. How do I show that?  Please help! Sauron's sight details? I am aware that the "Eye of Sauron" in Peter Jackson's movies was not Sauron's actual physical form.  What the answers at '' don't indicate is, despite this, what abilities of vision Sauron may have had.  Did he have the ability to see from afar? If so, what are his limitations or details of this power? Is it some kind of Elvish-sight like Legolas but stronger? Was it a magical power? Or perhaps with the help of the Palantir (I'm not sure of this last one because I dont remember if the Palantir could only see what other Palantíri were seeing or anything he wanted)? Let $L$ be a finitely generated free module over a commutative ring $A$. Set $n=\operatorname{rank} L$. Let $x_1,\dots,x_m$ be generators of $L$. Then $m \ge n$? If $m = n$, then is $x_1,\dots,x_m$ a basis of $L$? If $\gcd(a,b)=1$, $\gcd(a,y)=1$ and $\gcd(b,x)=1$ then prove that $ax+by$ is prime to $ab$.  I tried assuming Diophantine Equations for all the relations and representing everything in terms of $a$, but that didn't lead me anywhere. There must be an intuitive and elegant approach right? Agreement in "[Singular Noun] Is/Are [Plural Noun]"?