Mr. Dursley arrived in the Grunnings parking lot. We use "at" for a specific place. So why "in" used in this sentence?
When do we use "at" and "in" with "arrive" talking about place, not time?
$S^2 \times S^2$ is diffeomophic to $G_2(\mathbb{R}^4)$ $G_2(\mathbb{R}^4)$ is the Grassmannian manifold of two-dimensional subspaces of $\mathbb{R}^4$. I would like a detailed proof. Can it be done explicitly? I mean, showing the map and checking its differentiability using charts? Does ASP.NET MVC use the regular toolbox controls? Inner product on $C(\mathbb R)$ With axiom of choice it is possible to construct an inner product on $C(\mathbb R)$. My question is, is it possible to explicitly construct an inner product on $C(\mathbb R)$? I.e. to give a closed formula to calculate the inner product? I know it is straight-forward to write down a scalar product using a Hamel basis. This is not the answer I am looking for. This question came to me, when a student asked me in the lecture today 'whether there are vector spaces without inner products'. So I tried to find scalar produces for function spaces. I think I managed to write one down for $L^1((0,1))$. But I failed to construct one for $C(\mathbb R)$. When and why is the [featured] tag removed by the Community user? Buying from AliExpress I've notice that the business to business site AliBaba and it's more consumer friendly AliExpress website have a number of bike components for sale. They tend to be complete component groupsets and the like but are some quite reasonable deals, especially if you are building a bike from scratch.  I've used AliExpress before to purchase electronic components, which have been good quality and are what they say on the tin.   Has anyone used AliExpress to buy bike components and have they had any trouble? (quality, counterfeit etc.)  One benefit is that your money is held in escrow until you receive the item and aprove the payment so some safety. Does "Erase Disk and Install Ubuntu" replace the default boot screen? I am new here. And I may replace a laptop's hard disk with Ubuntu. It's an HP Compaq notebook which has Windows 7 installed. So If I choose that install option, will it replace the screen that says F12 for boot options [actually it was esc]. I am just asking. I know that choosing that options erases your other partitions. I am assuming it gets replaced by grub2? Questions can be protected (i.e. marked "highly active").   What does it mean for a question to be protected? Why are some questions protected? Who can protect and unprotect questions? Who can answer a protected question? How are protected questions displayed? When should I protect or unprotect a question? When does the Community user protect a question? How do I pass have a Javascript script request a PHP page and pass data to it? How do I then have the PHP script pass data back to the Javascript script?  client.js:  data = {tohex: 4919, sum: [1, 3, 5]}; // how would this script pass data to server.php and access the response?   server.php:  $tohex = ... ; // How would this be set to data.tohex? $sum = ...; // How would this be set to data.sum? // How would this be sent to client.js? array(base_convert($tohex, 16), array_sum($sum)) I have already proved Triangle Inequality $\lVert v+w\rVert \le \lVert v \rVert+\lVert w \rVert$ in vector space $\mathbb{R}^n$. However, I'm having difficulties proving the Reverse Triangle Inequality in the vector space.  I have started with   $$\lVert v \rVert^2=v \cdot v-w\cdot w+w\cdot w \le \lVert v-w \rVert+\lVert w \rVert^2$$ $$\lVert w \rVert^2=w \cdot w-v\cdot v+v\cdot v \le \lVert w-v \rVert+\lVert v \rVert^2$$  based on the Triangle Inequality but I don't know if it is correct. If this is right, I think I can go on with proving it the same way I would the normal Reverse Triangle Inequality. Probability of correctly guessing 12 zodiac signs without replacement I just watched  where someone assigns a zodiac sign to twelve different individuals. There is one of each sign.  My question, Guessing at random, (this was not quite at random), what number of people are you expected to guess correctly? And with what probability are you expected to guess a given number of people correctly?  i.e. What are the odds of getting 1, 2, 3, 4, etc. correct. Consider a database table holding names, with three rows:  Peter Paul Mary   Is there an easy way to turn this into a single string of Peter, Paul, Mary? Gaussian integral evaluation Asked a question to evaluate the Gaussian Integral,  $$\dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty x^2 \exp(-x^2/2) dx  $$  using the the following approximation,  $J=\Bbb E[X^2] \sim J_N = 1/N \sum_1^NX_n^2 $  where $x_n \sim N(0, 1)$  Examin $J_N$ for different values of $N$ and plot an error graph (difference between exact and numerical values)  I can simulate and generate the approximations but I am stuck at getting the exact value to compare to. As far as I am aware there are no analytical solutions for gaussian integrals. The other way I can generate comparable numbers is to go use numerical integration techniques but those are approximate values too.  Any help will be greatly appreciated! Difference between == and === in JavaScript Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$ Prove the following statement $S(n)$ for $n\ge1$:  $$\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$$  To prove the basis, I substitute $1$ for $n$ in $S(n)$:  $$\sum_{i=1}^11^3=1=\frac{1^2(2)^2}{4}$$  Great. For the inductive step, I assume $S(n)$ to be true and prove $S(n+1)$:  $$\sum_{i=1}^{n+1}i^3=\frac{(n+1)^2(n+2)^2}{4}$$  Considering the sum on the left side:  $$\sum_{i=1}^{n+1}i^3=\sum_{i=1}^ni^3+(n+1)^3$$  I make use of $S(n)$ by substituting its right side for $\sum_{i=1}^ni^3$:  $$\sum_{i=1}^{n+1}i^3=\frac{n^2(n+1)^2}{4}+(n+1)^3$$  This is where I get a little lost. I think I expand the equation to be  $$=\frac{(n^4+2n^3+n^2)}{4}+(n+1)^3$$  but I'm not totally confident about that. Can anyone provide some guidance? Automatically redirect anonymous user from unanswered duplicate question to corresponding answered version Duplicate questions that use different words to describe the same problem are important for people to find the content via search engines. But the user experience when an anonymous user arrives at a duplicate is less than ideal.  They have to notice the duplicate link and understand that the answer they are looking for is likely behind that link. I don't think this is all that obvious to someone without experience with an SE site.  I propose to just redirect users to the duplicate automatically if the closed duplicate has zero answers. In that case they need to follow the link to the duplicate in any case, so we might as well redirect them to it in the first place.  I could imagine this being problematic with search engines, as we would be displaying different content than what the search engines see. If that is too problematic I would suggest to show a very obvious link to the duplicate in the answers section for anonymous users. In Harry Potter, Who is Merlin? In Harry Potter, who is Merlin and what is his role in wizarding history?  His contribution to the wizarding history is apparently known from the fact that some special honors are named after him, such as the Order of Merlin (First Class, Second Class and so on...). Did he live around the time of the four founders of Hogwarts? In the current era of cricket, we see batsmen using switch hits to play a ball. That is, while playing a ball they change their batting side (left to right handed and vice versa) just before the ball comes to them.  My question is, can a bowler bowl a switch ball? Meaning, can a left hand bowler bowl right handed to a batsman? If it is not permitted, then is it not unfair to the bowlers that batsmen can change their batting style but not the bowlers? Or are there are any rules against a switch hit? What to do with cross-site duplicates? What's the right course of action for cross-site duplicates?  There are two flavors:   A question is posted which belongs on another site and ends up already having been asked there. I can't find an example right now, but it's bound to happen soon if it hasn't already. The same question is posted in two places at the same time. Take  for example, which is a duplicate of , posted within a half hour of each other by the same user.   Possible actions:   Move it to the proper site and close it once it gets there as a duplicate. Provide a way to close it right away as a cross-site duplicate, without moving it first. Flag it and let a moderator merge them. Close it as "no longer relevant" or something else.   Additionally, flavor 2 is a bit of a special case; perhaps the question really spans both sites in terms of content and intended audience. Should multiple-postings in this case be allowed? If so, should they somehow be linked to each other? If not, how do we avoid a migration war? The Half-Elven and their choices What data type to use for hashed password field and what length? I'm not sure how password hashing works (will be implementing it later), but need to create database schema now.  I'm thinking of limiting passwords to 4-20 characters, but as I understand after encrypting hash string will be of different length.  So, how to store these passwords in the database?