Resolving nested dependent summation $ \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} j $
Getting a closed form from $\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1$
How to install Eclipse? I used sculpting on a plane, and things looks great. However, when I press P to play, the whole ground become a mess. I can see through it in various places, and there are also nasty looking black edges everywhere.  another problem, I made a UV Sphere to display the sky, but when I press , P to play it disappears! I have a subomain which I do not want to be listed in any search engine result. Let say I have:   http://www.example.com http://child.example.com   How can I hide all URLs of child.example.com domain which are currently showing in the search engine results? Is it possible to have site-specific badges? AF mirror not flipping on Canon 350D My Canon 350D is no longer raising the AF mirror when it takes a photo.  Instead, it looks like this when the mirror flips up.   This results in a photo that is half black due to the sensor being obscured.  This camera is 9 years old and has never been dropped, mishandled or subjected to harsh temperature or humidity.  Is there any common maintenance I can do to get this thing swinging properly again? Purpose of else and finally in exception handling My computer does not store the history of the entries entered in RUN (WIN+R).   Thus, how to enable Windows to store the history? How do I change my login provider(s)? I would like to compare the contents of a couple of collections in my Equals method.  I have a Dictionary and an IList.  Is there a built-in method to do this?  Edited: I want to compare two Dictionaries and two ILists, so I think what equality means is clear - if the two dictionaries contain the same keys mapped to the same values, then they're equal. How can I prove that the set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable?  Edit:  This answer came to mind.  Is it correct?  This answer just came to mind.  By contradiction suppose the set is $\{f_n\}_{n \in \mathbb{N}}$.  Define the function $f: \mathbb{N} \to \{0,1\}$ by $f(n) \ne f_n(n)$.  Then $f \notin\{f_n\}_{n \in \mathbb{N}}$. I would really like to see all of my questions and answers on the profile page, even if some of them were deleted, and I don't have enough rep to see them on the site. (Note that some questions are  after 30 days or 1 year, and the author might be oblivious about what happened.)   , the &quot;deleted recent [questions/answers]&quot; links in one's own user profile now list all respective posts that were deleted in the past 60 days, rather than just those that were originally posted in the last 60 days. , users with the &quot;access to moderator tools&quot; privilege (10k+ on designed sites, 2k+ on beta and non-designed sites) can use the search operator deleted:1 to see all their own deleted posts. , deleted questions and answers that were posted in the past 60 days, can be seen using the &quot;deleted recent questions&quot; and &quot;deleted recent answers&quot; links on the questions and answers tabs (at the bottom of the tab) in your user profile. For older posts: as of 2013-04-23 you can  if you already have a link to them, but they still aren't linked from the user profiles (not even just for you). Nor do you see inbox notifications for comments on them if you come back a few hours later. You have to have thought to bookmark your question, or you have to go digging for it in your browser history. What should I do if a user posts sensitive information as part of a question or answer? Problems with Curve Modifier Every time I think I have the curve modifier mastered, it throws me a another curve ball (no pun intended).  Why in  can I not get the lights to follow the spiral curve? When the modifier is activated, the lights are stretched and distorted along the curve.  I created the curve by converting it from a mesh (string of verts). Why does it not display like a normal Bezier? Is that related to my problem?  I am following all the "rules" that I am aware of, as detailed here: . The curve seems to run in the correct direction. The origin point is at the first point of the curve. The curve modifier on the lights is set to -Y which is the local orientation of the curve's direction.  What am I missing? General relativity is expressed in terms of differential geometry, which allows you to do interesting things with the coordinates: multiple coordinates may refer to a single point, eg. the  has whole lines at the top and the bottom that correspond to the North and South poles, respectively; or a single coordinate may refer to multiple points, for example by using inverse geometry the origin refers to all the points infinitely far away.  So, is the singularity just a single point in the curved spacetime, or can it be a more extended object, described by a single coordinate? How many times to roll a die before getting two consecutive sixes? Basically, on average, how many times do you have to roll a fair six-sided die before getting two consecutive sixes? Example of the equality of an inequality This question is related to Daniel Fischer's answer . Suppose $f$ is a real $C^{1}$ function on $[0, 1]$ such that $f(0) = 0$ and $\int_{0}^{1}f'(x)^{2}\, dx \leq 1$. Then (essentially by Cauchy-Schwarz), we have $\left|\int_{0}^{1}f(x)\, dx\right| \leq 2/3$ as Daniel Fischer stated in his answer.  My question is: Is there an example of a function such that we have equality in $\left|\int_{0}^{1}f(x)\, dx\right| \leq 2/3$? I was thinking of maybe a smoothed version of a function which takes the value $0$ on $[0, 1/2)$ and $4/3$ on $(1/2, 1]$ but that seems complicated to construct, moreover, the derivative in some neighbourhood of $1/2$ might be hard to control. java.lang.RuntimeException Cannot find FacesContext A word for someone who you share a half-sibling with This is probably best explained with an example.  Alice and Bob have a child, Erin. Bob and Carol have a child, Frank. Carol and Dan have a child, Grace. Clearly, Erin and Frank are half-siblings, because of their father, Bob. Similarly, Grace and Frank share their mother, Carol.  Erin and Grace have no parents in common, and are thus not half-siblings. However, is there a term for their relation? What are the origins of the word orientated?  As far as I know, the correct spelling is oriented and orientated is not an alternative spelling but an error that is in common use.  Is it for example more commonly used in a certain country or by a certain people? Is there a reason for people choosing to say or write orientated instead of oriented? Derivative of the product of operators and Derivative of exponential I'm asked to show that  $$\frac{d(\hat{A}\hat{B})}{d\lambda} ~=~ \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$$ With $\lambda$ a continuous parameter. Should I use the definition $$\frac{d\hat{A}}{d\lambda} ~=~ \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon) - \hat{A}(\lambda)}{\epsilon}$$ applied to $\hat{A}\hat{B}$ like $$\frac{d(\hat{A}\hat{B})}{d\lambda} ~=~ \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon)\hat{B}(\lambda + \epsilon) - \hat{A}(\lambda)\hat{B}(\lambda)}{\epsilon}$$ and do some algebra to get the RHS of the first equation, or I'm missing something? Another interesting derivative to pay attention to is: $$\frac{d}{d\lambda}\exp(\hat{A}(\lambda) )~?$$