$\lim_{n\rightarrow\infty} H_n-H_{\frac{n}{2}}$
The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$
Gauge symmetries and elementary particles The  (disclaimer: I don't know this wikipedia entry) is usually mentioned as the reason why gravitons may not be composite particles. I do understand the proof of the theorem, but not the previous conclusion.   The theorem states that in an interacting and Poincare invariant quantum field theory, there are not massless, spin-2 particles unless there exists a gauge symmetry — which makes the energy-momentum tensor non-covariant (actually covariant up to a gauge transformation) under Lorentz transformations in the Fock space. So the immediate conclusion of the theorem is that the existence of a massless, spin-2 particle (like a graviton) requires linearized diffeomorphisms.  My question is: why do linearized diffeomorphisms imply that gravitons are elementary particles? Or, more in general, why  the particle corresponding to a gauge field must be elementary (I know that a gauge symmetry must be exact, but why this implies that the corresponding particle must be elementary?). What are the most suitable filter holders for my Samyang 14mm autofocus prime lens? I've looked into the ones from Haida and Nisi which are both suitable, though they're quite pricey for me currently - does anybody know of any cheaper possible alternatives? I understand there's the 160mm specifically made for use with this lens, though obtaining filters for it is difficult and limited. I'm hoping for something which may be cheaper than both Haida and Nisi, and still suitable for the 14mm lens. Thanks! How to favorite a question from Android app? If sup A $\lt$ sup B show that an element of $B$ is an upper bound of $A$ (a) If sup A &lt; sup B, show that there exists an element of $b \in B$ that is an upper bound for $A$.  I have argued that if sup A $\lt$ sup B, then choose an $\epsilon&gt;0$ such that sup A +$\epsilon \in B$. Since $a \le $ sup A for all $a \in A$, it follows $a \lt $ sup A + $\epsilon$ hence sup A + $\epsilon $ is an upper bound of A as well as an element of B.  (b) Give an example to show that this is not always the case if we only assume sup A $\le$ sup B.  I am having trouble extending my argument into an example which makes me think that it may not be correct.   Question: Is my argument for (a) correct? If not how would one show this fact? Can you give an example asked for in (b)? Thanks in advance.  I need to understand how I am assuming what I am supposed to prove in my approach. Can someone elaborate on that please? The other question doesn't really address this issue. php execute a background process I need to execute a directory copy upon a user action, but the directories are quite large, so I would like to be able to perform such an action without the user being aware of the time it takes for the copy to complete.  Any suggestions would be much appreciated. Terminal, how to quit --More-- list I have Opportunity-lookup field on Opportunity (self),  Scenario:   Opportunity A if select Opportunity B using lookup then Opportunity B's same lookup field automatically populate with Opportunity A.   I tried to achieve this using trigger - before insert/update but its failing as:  System.DmlException: Update failed. First exception on row 0 with id 006G000000DHaBlIAL; first error: SELF_REFERENCE_FROM_TRIGGER, Object (id = 006G000000DHaBl) is currently in trigger Oppy_before_INSUPD, therefore it cannot recursively update itself: [] Find the value of : $\cos x \cos 2x...\cos 999x$ given that $x=\frac {2\pi}{1999}$ Given $x=\frac {2\pi}{1999}$  Find the value of $$\cos x \cos 2x \cos 3x ...\cos 999x$$  So I tried expanding $\sin {2000x}=2\sin 1000x \cos 1000x$  Then rewriting  $\cos 1000x= \cos {(999x+x)}$   No luck so far. Possible Duplicate:          Please, look at the picture?    Which calculator shows true result and how to prove achieved result? I've noticed that if I downvote an answer, the record of the downvote is visible to me in my profile.  But when I view my profile from another browser that's not logged into Stack Overflow, I've noticed that I don't see the record of the downvote.  So I was wondering if the downvote record is visible only to me?  If so, is this done to protect downvoters from retaliation? How to push a copy of an object into array in PHP Let $A$ be the $\mathbb{C}$-subalgebra of $M_4(\mathbb C)$ consisting of matrices of the form   \begin{pmatrix} * &amp; * &amp; * &amp; *\\ * &amp; * &amp; * &amp;*\\ 0 &amp; 0 &amp; *&amp;0 \\ 0 &amp; 0&amp;* &amp;* \end{pmatrix}   Then how would I find the Jacobson radical $J(A)$ of $A$?    I don't really understand finding the Jacobson radical of $\mathbb{C}$-subalgebras.  Any help on this would be greatly appreciated, thank you! Controlling colors of points in ListPlot ASP.NET "special" tags Present perfect for past action with present effect If I seem tired, can I say: "I haven't slept last night"? If not, why have I been told that we use present perfect for actions that have present effects? Electrostatics and tv screens Truncate (not round) decimal places in SQL Server Thus is a picture of both of my hard-drives:    My C Drive and my D drive. I want to put Ubuntu onto my D Drive. Now my question here is if i do install onto my D Drive then how should I do it in order to retain use of the drive. I intend to partition about 60 gigs from it. Now after doing that I should still have about 830 gigs left. I want to still be able to use that in Windows. So how and will I be able to retain use of the drive after Installing? I have installed once before and I believe i did it wrong because not only could Windows not recognize my D drive but I also lost all the data on it.   My computer is a Lenovo Y700 It is a UEFI (I believe i got that right). I use (and have it much bigger) drive D: to store all my user data and games, but I have installed the Steam client on drive C:. When I was installing a first game (Warhammer 40000), I've chosen to place my library folder on drive D: and the game went there.  But now as I've got installed the second game (Portal), I can see that it was put on drive C: despite to the choice I've done previously. When I open Steam - Settings - Downloads + Cloud - Steam library folders, I can see both installation places in the list. Can I merge them into one?  I would even agree to use the default drive C: library (as I've replaced it with a symlink already). When I press to remove one library it says it can only remove empty ones. Quantum computers are known to be able to crack  a broad range of cryptographic algorithms which were previously thought to be solvable only by resources increasing exponentially with the bit size of the key. An example for that is .  But, as far I know, not all problems fall into this category. On , we can read     Researchers have developed a computer algorithm that doesn’t solve problems but instead creates them for the purpose of evaluating quantum computers.   Can we still expect a new cryptographic algorithm which will be hard to crack using even a quantum computer? For clarity: the question refers to specifically to the design of new algorithms.