Modulo 2 binary long division in European notation
Typesetting long division with tabular and overlining repeating part
Captchas not displaying in SO for me When I get a captcha on Stack Overflow, I don't see the image or the input box with Firefox. I originally had the "HTTPS Anywhere" extension installed but still have the same issue when it's disabled (This caused issues previously when Google's SSL certs were expired for an old domain IIRC).  The requests to Google seem to be never-ending; browsing to the URLs that are still loading results in some pages of JavaScript, but no errors as you'd hope.  The screenshot was taken after the page had been open for more than 3 minutes, and I'm not sure why the GETs are still running. The request popup visible in Firebug is for "GET challenge?k=...."  When browsing to the same captcha page in IE, there are no issues.  I'm aware this could be an issue with my Firefox configuration, as opposed to the site, but would appreciate it if someone could verify if it's just me, and suggest a fix.  EDIT   While visiting the Recaptcha site, I can see a Captcha with no problems If I right-click the problem URLs in FF and Open in new tab, the pages load fine. From the fact that FB omits IP Addresses for the problem ones, I assume that it's not even opening a connection (nor perhaps performing a DNS lookup) Other sites that use Recaptcha seem to work fine (I picked 3 at random from Google) $\sqrt{1-x^2}$ is not a rational function If $k$ is a field in which $1 + 1 \neq 0$, prove that $\sqrt{1-x^2}$ is not a rational function. Hint. Mimic the classical proof that $\sqrt{2}$ is irrational   It seems rather odd to discuss the square root function in the context of an arbitrary field, but here goes nothing!  Proof:   Suppose that $\sqrt{1-x^2}$ is a rational function in $k(x)$. Then there exist polynomials $p(x)$ and $q(x) \neq 0$ that are relatively prime with $\sqrt{1-x^2} = \frac{p(x)}{q(x)}$; clearly we may take $p(x) \neq 0$. Then $1-x^2 = \frac{p(x)^2}{q(x)^2}$ or $q(x)^2(1-x^2) = p(x)^2$, which says $q^2 | p^2$, and since $q | q^2$, we can infer $q | p^2$. Moreover, since $p|p^2$ and $(p,q)=1$, then $pq|p^2$ or $q|p$, which contradicts the fact that the polynomials are relatively prime. Hence $\sqrt{1-x^2}$ cannot be a rational function.    How does this sound? Aside from the problem of discussing $\sqrt{~~}$ in the context of an arbitrary field, I am worried about not using the fact that $1+1 \neq 0$, at least not explicitly. Where exactly is this assumed used, if at all?  EDIT:  Suppose that $q(x) =1$, and let $f(x) = a_n x^n + ... a_1 x + a_0$. Then $1-x^2 = f(x)^2$. If $x=0$, $1 = a^2_0$ and therefore $a_0$ must be a unit. Letting $x=1$ we get $0 = (a_n + ... + a_1 + a_0)^2$; and letting $x=-1$ we get $0 = (-a_n - ... - a_1  + a_0)^2$. Since we are working in a field, there can be no nonzero nilpotent which means that $a_n + ... + a_1 + a_0$ and $-a_n - ... - a_1  + a_0$ are both zero. Adding the two equations together yields $2a_0 = 0$, and since $1+1 \neq 0$, $a_0 = 0$ which contradicts the fact that $a_0$ is a unit.   How does this sound?  Another attemtpt:  If $f(x)^2 = 1-x^2$, then $2 = \deg(f^2) = 2 \deg (f)$ implies $\deg(f) = 1$. Yet if $x=-1$, we get $f(-1)^2 = 1 - 1 = 0$ and therefore $f(-1)=0$ since there are no nonzero nilpotent elements in a field; similarly, $f(1)=0$. Since $1+1 \neq 0$, $f$ has two distinct roots in $k$ yet is only a $1$-st degree polynomial, a contradiction. Nuclear Binding energy Volume control on asus zenbook 3 Using Ubuntu 16 LTS  I installed ubuntu on my new asus zenbook3 ux390. Nearly everything works fine except the volume controls don't work. The volume is either on or off. I can adjust volume in specific apps or websites (like youtube) successfully, but the internal volume control either wants to be muted or at 100%. {5,15,25,35} is a group under multiplication mod 40 Show that the set $\{5,15,25,35\}$ is a group under multiplication modulo $40$. What is the identity element of this group. Can you see any relationship between this group and $\mathbb U(8)$?   I am very stuck on this question and I think my knowledge of abstract algebra is my liability at the moment. Right now I'm most confused by the fact that I learned: An element of a group of this nature must be relatively prime with the working mod.  Obviously here this is not the case, so I don't really know where to begin.  Do I need a Cayley Table?  Thanks. Last Logon Time of All Users - RDP server not on a domain I am attempting to get a list of "last logon time" of "all users" on Windows Server 2012, but currently only know of how to list a single user login, which is:  net user username | findstr /B /C:"Last logon"  Any ideas?  *P.S. This servers only purpose is to host the RDP connections; not tied to a domain/AD. How to cut the non-overlapping part of 2 shapes in Illustrator CS6 I have 2 shapes (see below) a circle and a shape of the US. I want to remove the part of the US that falls outside the circle. How is this possible?    Thanks I have created some search entries in Google Chrome using 'Edit search engines'.  How can I share some of these entries with my colleagues? I submitted a manuscript last year to an IEEE journal, and the paper was accepted for publication. I recently submitted a new manuscript to the same journal, but, this time, the paper was rejected, and I was told that I should submit to another journal since my paper was not within the journal's scope. The two papers both target similar applications. Furthermore, the second paper uses some work in the first paper, making substantial improvements at solving a much more difficult problem. Based on the reviewer's comments (and my own personal assessment), I do not believe that the work presented in the second paper was "incremental" in nature. Far from it. Two of the three reviewers gave positive reviews. The third reviewer, however, mentioned the "outside of journal's scope" issue and the associate editor and editor-in-chief both sided with the 3rd reviewer, and the paper was stopped dead in its tracks.  I'm scratching my head trying to come to terms with how in the world my first paper could be accepted by this journal, but the second paper focused on the same application space and achieving substantial improvements on a much more difficult problem (backed up with measured results) could be rejected. Any ideas? How do I add a custom script on a specific page? I wanted to add custom JavaScript code on certain nodes. I have written this code in the html.tpl.php template, but this doesn't work. (donate is the path alias for a node.) &lt;?php if(arg(0) == 'DONATE'): ?&gt;   &lt;script&gt;     (function() {       // ...     );   &lt;/script&gt; &lt;?php endif; ?&gt; Regularization using methods such as Ridge, Lasso, ElasticNet is quite common for linear regression. I wanted to know the following: Are these methods applicable for logistic regression? If so, are there any differences in the way they need to be used for logistic regression? If these methods are not applicable, how does one regularize a logistic regression? Exception in thread "main" java.lang.ArrayIndexOutOfBoundsException What is the effect where some objects are a single bright color but the rest is black and white? There is an option to filter only one color (R,G,B,Y) in my camera. What I want to know is can this effect be achieved in photoshop? And what is this called? I tried googling monochrome and came up with nothing. My camera calls it part color. Why was Dumbledore worried by the cut in Harry's arm? In the Goblet of Fire, Harry is cut in the arm by Pettigrew. When Dumbledore learns it, he's extremely worried and wants to check Harry's arm.  Why?     When Harry told of Wormtail piercing his arm with the dagger, however, Sirius let out a vehement exclamation and Dumbledore stood up so quickly that Harry   started. Dumbledore walked around the desk and told Harry to stretch out his arm.   Harry showed them both the place where his robes were torn and the cut beneath   them.      "He said my blood would make him stronger than if he'd used someone else's,"   Harry told Dumbledore. "He said the protection my - my mother left in me - he'd   have it too. And he was right - he could touch me without hurting himself, he   touched my face."      For a fleeting instant, Harry thought he saw a gleam of something like triumph in   Dumbledore's eyes. But next second. Harry was sure he had imagined it, for when   Dumbledore had returned to his seat behind the desk, he looked as old and weary   as Harry had ever seen him.      "Very well," he said, sitting down again. "Voldemort has overcome that particular   barrier. Harry, continue, please.     Edit:  Plz explain this sentence specifically:     "Dumbledore walked around the desk and told Harry to stretch out his   arm." How do you define an arbitrary sort sequence for views? Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism? Let $V$ and $W$ be vector spaces (say over the reals). There is a linear injection $V^* \otimes W^* \to (V \otimes W)^*$ which sends $\sum_i f_i \otimes g_i \in V^* \otimes W^*$ to the unique functional in $(V \otimes W)^*$ sending $v \otimes w \mapsto \sum_i f_i(v) \cdot g_i(w)$ for all $(v,w) \in V \times W$. In the finite dimensional case it is easy to see this is an isomorphism by comparing dimensions on both sides. I'm aware that this is not an isomorphism in the infinite-dimensional case (see Relation with the dual space in the  on tensor products) but I must admit that I'm not sure why. Cardinal arithmetic does not seem help here and, even if it did, I would be much happier to see an explicit example of a functional in $(V \otimes W)^*$ outside the range of this map. I'm having trouble cooking one up myself.  Added: Let me elaborate on my comment about cardinal arithmetic. Suppose that $X$ and $Y$ are infinite dimensional vector spaces over a field $k$. I'm confident that $\dim (X \otimes Y) = \dim X \cdot \dim Y$ holds. It is clear that $|X^*| = |k|^{\dim X}$ since we may identify a functional on $X$ with a function from a basis for $X$ to $k$. Also I think I've convinced myself that if $\dim X \geq |k|$ then in fact $|X| = \dim X$ ie. the cardinality and dimension of a vector space agree when the dimension is larger than the cardinality of the ground field. Consequently, assuming say that $|k| \leq \dim X \leq \dim Y$ it seems we have $$ \dim(X \otimes Y)^* = |k|^{ \dim X \cdot \dim Y} = |k|^{ \dim Y} = \dim Y^* = \dim X^* \cdot \dim Y^* = \dim (X^* \otimes Y^*)$$  which implies that in fact $(X \otimes Y)^*$ and $X^* \otimes Y^*$ are isomorphic but, rather frustratingly, the obvious map does not do the job. Did I make a mistake here? If not, is it generally true (ie without making assumptions about $|k|$) that there exists some isomorphism $(X \otimes Y)^* \to X^* \otimes Y^*$? Can I kill an indestructible creature with -X/-X and sufficient damage marked on it? Can I hit a 4/4 indestructible creature with a  (3 damage) and give it -1/-1 with  to kill it? (Does it matter which way around I cast those?)  I read  here and it says, damage is removed in cleanup step, so it should be on the creature still after both spells.     702.12. Indestructible   702.12a Indestructible is a static ability.   702.12b A permanent with indestructible can’t be destroyed. Such permanents aren’t destroyed by lethal damage, and they ignore the state-based action that checks for lethal damage (see rule 704.5g).   702.12c Multiple instances of indestructible on the same permanent are redundant. I am a little at a loss.  A couple years before my mother-in-law passed away she signed her house over to my wife and her sister via a quitclaim deed.    After she passed we sold the house for $80,000. The money was used to pay off the mortgage, some other funeral expenses, and sent $15,000 to other family members.  All the money was dispersed at the closing of the house.  My wife and her sister each ended up with about $20,000.  How do we go about claiming this income?  Since there was a quitclaim deed I guess we technically owned the house for a few years prior to its sale.  How does that affect our tax and what is the basis?  Is the basis the selling price(fair market value) or something else?  How much does each sister claim for gains?  Sorry it this is a jumble of questions I am just trying to get this straightened out before tax is due. What is the correct way to create a single-instance WPF application? Using C# and WPF under .NET (rather than  or console), what is the correct way to create an application that can only be run as a single instance?  I know it has something to do with some mythical thing called a mutex, rarely can I find someone that bothers to stop and explain what one of these are.  The code needs to also inform the already-running instance that the user tried to start a second one, and maybe also pass any command-line arguments if any existed. Using customized Coordinate System in ArcGIS Desktop? I don't know so much about coordinate systems...  In my office we use to deal with spatial data coming from archaeological sites. Each site has its own x-y-z coordinate system (GCS). Three simple ortogonal cartesian axis. In the last years we have been managing this spatial data through GIS software (ArcGIS), without using specific coordinate system (just leave it as "undefined")  I'd like to know if there exisits any GCS designed to deal with such datasets using simple cartesian orthogonal axis, without grid distortions of the typical GCS. In addition, I'd like to know if this system is suitable for using it in an online mapping application.  By the way, we manage 2D (ArcMap) and 3D (ArcScene) environments and work with "mm" as length base unit.   If such a thing doesn't exists, maybe someone knows how to create it.