Find the recurrence equation for coloring a 1 by n chessboard
Need to find the recurrence equation for coloring a 1 by n chessboard
Curve Bevel function, scale? Latest earned tag badge doesn't update when I earn a new one When you earn one tag badge, and then don't click "Track the next one" in the tag badge progress tracking box for a while, and then earn a second tag badge, that box continues to show "Congratulations, you earned [the same badge as before]", rather than updating to show that you've actually earned another tag badge since then.  For instance, right now  looks like this:    Even though I  a few weeks ago and then  yesterday, my profile hasn't updated to show the new achievement.  I also didn't get the "cipher" tag badge until after the 3am watershed, even though tag badges usually appear as soon as I visit my profile after earning them (for instance, I earned the "tags" tag badge on meta.SFF.SE today the very instant I clicked on my own profile there after having become eligible for the badge). Even when I went to look at the list of my progress towards different tag badges on Puzzling, which I thought would jog the system into realising I'd actually earned the badge, "cipher" still showed up as 99/100 score even after I'd gone well past 100, until last night when the badge was finally awarded and "cipher" disappeared from my list of badges to track.  TL;DR: if I don't click "Track the next one", my tag badges update slower than they should.  It's a very unimportant bug, but there must be more people than just me who like to see the "Congratulations" thing there for a while and rest on their laurels before moving on to track their progress towards the next milestone. Show default image when no uploaded image Is there a phrase that means sleeping with someone without sex? Where to start looking up if there's something wrong with cck field variables? in short: Something in my zen-based custom theme is keeping a couple of cck fields from being display on my page. Basically my question is where to get access to the variables that are supposed to contain the html for the cck fields (which are missing when I was looking them in firebug). Below I explain why I think it's looking up the variables that might bring me closer to a solution  So I made this custom node type (events) with cck fields added. Also made some content using those fields. Added some css and it all displayed fine. Later the taxonomy terms below the title and some metadata bugged me so I followed along with a 'theming cookbook' pdf book that taught me how to make a node-events.tpl.php and simply deleted php variables like $terms. This worked great.     Obviously this also means that all this time, the view permissions of those fields have been checked and also that the display options under 'display fields' have been set to standard all the time. I also ruled out that the issue could be anything else than theme related as my cck fields do show up in a stock Drupal theme. The theme for my site is a zen-based custom theme.   So then even later, after I made a view for this node type and started adding css to that view I noticed at some point that my image field and two other fields are missing. It's hard to tell if they disappeared because of things that I did when creating the view and adding css (which would surprise me) I'm writing a contribution for a double-blind submission system.   Since I would like to supply more data than I could show on a small number of pages, I'm looking for a way to reference supplemental data in my manuscript, such as software scripts, figures or databases.  I could upload on GitHub, RPubs, Figshare and Zenodo, but all of these would expose my real user name (and I don't feel like making a throwaway e-mail account with a new repository for every anonymous publication that I intend to write).  Is there any way to get the above services to do what I want? Or is there a similar service that would allow anonymous hosting of research data?   It would be even better if I could later (e.g. after publication) declassify the information, with the link from the manuscript still being intact. How to make Illustrator not to lie about colors? An infinite series plus a continued fraction by Ramanujan Here is a famous problem posed by Ramanujan     Show that $$\left(1 + \frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \cdots\right) + \left(\cfrac{1}{1+}\cfrac{1}{1+}\cfrac{2}{1+}\cfrac{3}{1+}\cfrac{4}{1+\cdots}\right) = \sqrt{\frac{\pi e}{2}}$$   The first series seems vaguely familiar if we consider the function $$f(x) = x + \frac{x^{3}}{1\cdot 3} + \frac{x^{5}}{1\cdot 3\cdot 5} + \cdots$$ and note that $$f'(x) = 1 + xf(x)$$ so that $y = f(x)$ satisfies the differential equation $$\frac{dy}{dx} - xy = 1, y(0) = 0$$ The integrating factor here comes to be $e^{-x^{2}/2}$ so that $$ye^{-x^{2}/2} = \int_{0}^{x}e^{-t^{2}/2}\,dt$$ and hence $$f(x) = e^{x^{2}/2}\int_{0}^{x}e^{-t^{2}/2}\,dt$$ Thus the sum of the first series is $$f(1) = \sqrt{e}\int_{0}^{1}e^{-t^{2}/2}\,dt$$ But I have no idea about the continued fraction and still more I am not able to figure out how it would simplify to $\sqrt{\pi e/2}$ at the end.  Please provide any hints or suggestions.  Update: We have $$\begin{aligned}f(1) &amp;= \sqrt{e}\int_{0}^{1}e^{-t^{2}/2}\,dt = \sqrt{e}\int_{0}^{\infty}e^{-t^{2}/2}\,dt - \sqrt{e}\int_{1}^{\infty}e^{-t^{2}/2}\,dt\\ &amp;= \sqrt{\frac{\pi e}{2}} - \sqrt{e}\int_{1}^{\infty}e^{-t^{2}/2}\,dt\end{aligned}$$ and hence we finally need to establish $$\sqrt{e}\int_{1}^{\infty}e^{-t^{2}/2}\,dt = \cfrac{1}{1+}\cfrac{1}{1+}\cfrac{2}{1+}\cfrac{3}{1+}\cfrac{4}{1+\cdots}$$ On further searching in Ramanujan's Collected Papers I found the following formula $$\int_{0}^{a}e^{-x^{2}}\,dx = \frac{\sqrt{\pi}}{2} - \cfrac{e^{-a^{2}}}{2a+}\cfrac{1}{a+}\cfrac{2}{2a+}\cfrac{3}{a+}\cfrac{4}{2a+\cdots}$$ and it seems helpful here. But unfortunately proving this formula seems to be another big challenge for me. How to view an encrypted view or stored procedure I am working on a third party database.  When I try to view the definition of a view by right clicking, CREATE TO and then to NEW QUERY EDIT WINDOW, I am getting an error:     This property may not exist for this object or may not be retrievable   due to insufficient access rights. The text is encrypted. My PC has a build-in VT1828S chip. The connection between PC and monitor (which has built-in speakers) is through an HDMI cable.  Audio preferences shows 2 devices for audio output: analog stereo and RS880 Audio Device [Radeon HD 4200] digital stereo (HDMI).  When I plug a headphone in the rear jack and I choose the former device, everything works. When I choose the latter device, instead, no sound is heard, and video playback is too fast. It is as if video frames want to synchronise with the audio timing, but the audio timing information is wrong.  HDMI audio worked perfectly with Ubuntu 11.04, but stopped after 11.10 upgrade. What are instructions on how to download videos from YouTube?    Notice: Terms of Service Violation  Please be aware that by following any of the answers below, you will be violating YouTube's . In particular, from Section 5.B.:     Content is provided to you AS IS. You may access Content for your information and personal use solely as intended through the provided functionality of the Service and as permitted under these Terms of Service. You shall not download any Content unless you see a “download” or similar link displayed by YouTube on the Service for that Content. You shall not copy, reproduce, distribute, transmit, broadcast, display, sell, license, or otherwise exploit any Content for any other purposes without the prior written consent of YouTube or the respective licensors of the Content. YouTube and its licensors reserve all rights not expressly granted in and to the Service and the Content. Consider the real-valued function  $f(x)=x^r$ where $r$ is a real number.  (1) For what values of $x$ and $r$, is this function differentiable?  (2) How do you prove the power rule, when $f(x)$ is differentiable?  It would be nice if the proof is accessible to a 12th grade student.  EDIT: Perhaps, I have not been very clear about my question.  I am not satisfied with the "textbook proofs" I have seen.  Please established the existence (of the derivative) first before proving the power rule.  That is, please consider first question first.  This is the reason why I call the question a mathematical engineering question.  (If we can we bring high level math down to Grade 12 level, without loosing precision, then I call it good mathematical engineering.) I have a stupid self-answered question. Should I accept my own answer? was genuine at the time, but I realised that  gave the right advice, but required me to restart my Linux (which wasn't mentioned in the solution).  It's a bit of a  question and  points to me not deleting it.  But it seems hugely self-aggrandizing to accept my own answer. Should I? About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$ Why does this steam engine fail to run? Is it hard to transfer contacts from one iPhone to another iPhone? I just bought a new iPhone 6s, and want to move all the contacts from the old one to the new one. Can iTunes or iCloud help me do this, or are there any other ways? "Contact us"... and then what? So I had a dispute with the moderators at a specific SE site. As a result, I got suspended there. As per SE guidelines, I pressed the "Contact us" button and wrote to the higher powers to appeal.  That was 5-6 weeks ago. There was zero feedback, neither positive, nor negative, not even "it's in progress, please wait".  So what is this "contact us" thing supposed to do? Is it a placebo to make me feel better while being routed to dev/null?  What is the normal time it takes to process a "contact us" message?  Notes:   As far as I remember, I did not contact the team before. I'm a longtime user of multiple SE sites and never had any problems except for that one site. I'm pretty sure my form was submitted. I didn't make screenshots or anything, because I did not think I would need them in a post like this. I didn't expect a super-fast reply either, so I wasn't wondering after a week or two. Does the universe have an edge/boundary/barrier? If yes, what is at the edge? My question related kind of to asking what the shape of the universe is. Say a hypothetical alien civilization built an faster-than-light spaceship. If they keep flying would they end up where they began? ie space itself is curved...or would they hit some invisible barrier? What is duplicate content and how can I avoid being penalized for it on my site? In "Knots and Primes: An Introduction to Arithmetic Topology", the author uses the following proposition   Let $h: Y \to X$ be a covering. For any path $\gamma : [0,1] \to X$ and any $y \in h^{-1}(x) (x = \gamma(0))$, there exists a unique lift $\hat{\gamma} : [0,1] \to Y$ of $\gamma$ with $\hat{\gamma}(0) = y$. Furthermore, for any homotopy $\gamma_t (t \in [0,1])$ of $\gamma$ with $\gamma_t = \gamma(0)$ and $\gamma_t(1) = \gamma(1)$, there exists a unique lift of $\hat{\gamma_t}$ such that $\hat{\gamma_t}$ is the homotopy of $\hat{\gamma}$ with $\hat{\gamma_t}(0) = \hat{\gamma}(0)$ and $\hat{\gamma_t}(1) = \hat{\gamma}(1)$.  The author then follows with; "In the following, we assume that any covering space is connected. By the preceding proposition, the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$."   I am not sure why this results is true. This is my attempt at explaining it to myself. Take two different $x_1, x_2$ and their fibers $h^{-1}(x_1), h^{-1}(x_2)$ such that $y_1 \in h^{-1}(x_1)$ and $y_2 \in h^{-1}(x_2)$. Take two paths $\gamma_1, \gamma_2$ such that $\gamma_1(0) = x_1$ and $\gamma_2(0) = x_2$. Then we get two lifts $\hat{\gamma_1}, \hat{\gamma_2}$ with $\hat{\gamma_1}(0) = y_1$ and $\hat{\gamma_2}(0) = y_2$. Now, since our covering space is connected we can continuously deform $\hat{\gamma_1}$ into $\hat{\gamma_2}$ and conclude that every elements in $h^{-1}(x_1)$ is also in $h^{-1}(x_2)$ and vice versa. I feel that this is wrong but cannot figure out the right way to see this. Any help would be appreciated.