For someone not native to English, can you give your simplest guide on how to choose when to use between "in", "on", and "at" (prepositions)?
Should I say "She is in the park" or "She is at the park"?
Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$ I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows,  Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$  Consider $\frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2$  We can manipulate this as follows $$\frac{1}{4}(||u+v||^2 - ||u-v||^2) = \frac{1}{4}(||u+v||- ||u-v||)(||u+v||+||u-v||)$$ Using the triangle inequality the above can be written as follows, $$...\leq \frac{1}{4}(||u|| +||v|| -||u||+||v||) (||u|| +||v|| +||u||-||v||) = \frac{1}{4} (4||u||||v||) = ||u||||v||$$  As such both $u\cdot v$ and $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$ are $\leq ||u|| ||v||$  I'm not sure whether I've proven this correctly because I ended up proving that both sides of the equation are less than or equal to something entirely different (the magnitude of v multiplied by the magnitude of u). Is this acceptable? I don't think it is, since if both can be less than or equal to the magnitudes multiplied then they do not necessarily have to be equal to each other. If this is the case, can someone give me a hint on how to move forward with this proof? Comma before "and" in compound sentences? I read this sentence on an educational website:     Now times have changed and you are ready for situations involving forces in two dimensions.   Shouldn't there be a comma before and, since the subject of the sentence has changed (things vs. you)?      Now times have changed, and you are ready for situations involving forces in two dimensions.   What is the exact comma rule for joining two clauses with and? For every set $A$, the empty set is a subset of $A$. The empty set is a set. Therefore, the empty set has a cardinality $\geq 1\ldots$ I have only recently been exposed to sets. According to , as seen on the first bullet mark of the link, $ \forall A: \emptyset \subseteq A$  Does this mean that $\emptyset$ is an element of all sets? (This is False, thank you those that answered)  Is the empty set also a set itself?  Assuming these statements are true, then the empty set therefore an element of the empty set. This does not sound right, please clarify for me. Thank you. A clarification in a limit While reading this  I do not understand that how can we claim that there exist a polynomial with the properties, even if it exist then how is it same for the function and the inverse. Dual Channel RAM vs. Quad Channel RAM I just bought a motherboard that says it supports Dual-Channel RAM. The RAM I'm looking at says it's a "Quad-Channel Kit" . Will it work with my motherboard?  EDIT: Thanks all for the replies. Here is some more info:    Here is the RAM I'm looking at:    It all looks compatible to me, I just was curious about the four channel thing. What does a Terminator do after its mission is accomplished? A Terminator is essentially given a mission or an objective ("Go back in time and kill Sarah Connor" or something of the like).  What does a Terminator do once its mission is accomplished?  We see in Terminator 2: Judgment Day (the live-action movie) that the  (played by Arnold Terminator Schwarzenegger) completed its mission of protecting the Connors, and then sort-of just melted itself into nothingness (presumably to protect the world from discovering Terminators again).  But what would the  or the  (in Terminator 3) have done?   Are they given multiple directives ("Go kill Sarah Connor, then go kill Robert Baratheon, and then have a nice hot bath; you've earned it")?  This question is related, but not a dupe: Is hyperref really the best way to add metadata to a TeX file? So I know one of hyperref's features is that it can add metadata to you PDFs. This is a nice feature and if you're putting stuff online, it's good to include metadata.  But is the hyperref package really the best way to do it? I mean, if you're using hyperref anyway, that's fine. But if you don't want hyperrefs in your document, it seems weird to include the package, and then turn off pretty much all its functionality. And given that hyperref is a shorthand for "package conflict nightmare", is seems insane that you should have to deal with it just for the sake of some metadata.  So I have two questions.   Is there a good reason that hyperref includes this functionality which is apparently unrelated to its primary purpose? Is there a better (smaller, less conflicty, easier...) option to include metadata in my PDFs? Avoiding Mixed DML (setup/non-setup) Approach I need an approach to allow delegated admins (i.e., non-Administators) to add members to Public Groups and Collaboration Groups (the GroupMember and CollaborationGroupMember objects).  I added a custom object (UserGroupRequest__c) to capture which Users should be added/removed to which Groups.    I've also built a batchable class to process the requests.  However, GroupMember is a Setup object, and CollaborationGroupMember is not.  So this doesn't work if I combine the operations in a single batchable class.  Should I instead be using 2 batch classes, calling one from the finish method of the other?    Any recommendations on how best to proceed?  Am I way off base?    Would a trigger on the Request object perhaps be better? I'm guessing that there may be a permission problem if this is attempted via a trigger as the delegated admins won't be able to add users to public Groups.  Also, I'm guessing the trigger approach would run into the same mixed DML issue - on Insert of the Custom object, I would need to Insert/Delete from GroupMember and from CollaborationGroupMember, and then perform a 2nd update on the Custom object to indicate that processing was completed... How to proof that $\forall n \ge 1: \sum_{i=1}^n (i(i+1))^{-1} = n(n+1)^{-1}$ using mathematical induction So for a better presentation:   I need to prove this using mathematical induction, so i started with a base of $n=1$  Since $i$ starts at $i = 1$ I got this:  $$\frac{1}{i(i+1)} = \frac{n}{n+1}$$ $$\frac{1}{2} = \frac{1}{2}$$  So since this was true I went to the induction part: I chose $n = k$ and assumed this was true:  $$\frac{1}{i(i+1)} = \frac{n}{n+1}$$ $$\frac{1}{i(i+1)} = \frac{k}{k+1}$$  but I got stuck here since I didn't know if I needed to increment $i$ or not and because I didn't know if I needed to add the summation symbol or not and needed to do a few things with it. Is there a way to conveniently define a C-like structure in Python? I'm tired of writing stuff like:  class MyStruct():     def __init__(self, field1, field2, field3):         self.field1 = field1         self.field2 = field2         self.field3 = field3 Loading OpenStreetMap data in QGIS? In QGIS 1.8 I used the openstreetmap plugin to request data from the server, or open a .osm XML file, which produced 3 layers (polygons, polylines and points) in QGIS.  This was great (give or take the 64bit bug which highly affects the freshly mapped area I work on), but how do I do that in QGIS 2.x?   I can't find any updated documentation about it.  In the vector menu, I've tried importing from osm servers, I get a .osm file, but then I'm stuck. There's a menu option for loading an XML file, but it produces a spatialite db, which apparently only has non-geometry tables when I try to add it as a spatialite layer. If I open the attribute table, I can see node ids, ways and so on, but I can't figure out how to actually make use of them.  I've also tried adding a vector layer from a .osm XML file. I get a gpsbabel window that shows nothing, then finally an invalid data source message in QGIS. Convert date to another timezone in JavaScript I just wrote code for this, and I got it a little bit ugly. Can you help me to fix this. And how to change those nands for not, like it is on nand 3,4,5? Please help!  PS. What I mean by these nands for not or whatever this is Entire function bounded by a polynomial is a polynomial Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k&gt;0$ is constant. Show that $f$ is a polynomial of degree at most $n$. Fast way of finding lines in one file that are not in another? I want to install the latest version of Eclipse but the Ubuntu Software Centre contains an older version.  Is there a PPA or some other way to install latest Eclipse? Please describe the steps for full installation. split file lines based on first field How Do I get into the castle walls? Automatically Locate Included Images Do you think it's possible to include images into LaTeX without specifying the exact path to them. So instead of:  \includegraphics{path/to/image/myfig.pdf}   It would be:  \includegraphics{myfig.pdf}   While the image would still reside in the path/to/image/ directory. Basically, I'm asking if there is a macro that can recursively search for the image.  An alternative for me is to write a script outside of LaTeX that determines all the images that are included, performs the recursive search and copies those files temporarily in the same directory as the .tex file. Short story: rocket fuel which oxidizes iron destroys life on earth I'm looking for the title of a short science fiction story, probably published in a Hugo winner collection 30 years ago. In the story, scientists have developed an extremely efficient rocket fuel that has one dangerous side-effect: it oxidizes iron. A man takes off in a rocket powered by this fuel but returns to find the earth completely devoid of life. His subsequent investigation reveals that a single drop of the fuel began a domino effect, as the substance oxidized all iron it came into contact with. In the end, the man is dying.  He drags himself to the ocean, believing that if he can get his living cells into the water they will restart the primordial soup.