How to remove duplicates only if consecutive in a string?
Removing elements that have consecutive duplicates
Does the hom-tensor adjunction hold for $O_X$ modules also? With sheaf hom and sheaf tensor product, the statement would consist of a natural transformation $Hom_O (M \otimes_O N, K)\cong_{nat} Hom_O(M, Hom_O(N, K))$, $O = O_X$ is the structure sheaf and $M,N,K$ are sheaves of $O_X$ modules.  If true I think I can check this by hand, by describing sheaves in terms of compatible germs, defining this adjunction on the germs and checking that they glue together. It is easier in the quasi-coherent sheaves on a scheme case because then one can just work in an affine cover and the distinguished base.  I am little bit concerned that one might need a finite presentation hypothesis somewhere, in order for $Hom$ to localize well. ($Hom_{R[S^{-1}]}(M[S^{-1}], N[S^{-1}]) \cong Hom_R(M,N)[S^{-1}]$ needs $M$ finitely presented.) What are the norms in Ito isometry? Itō isometry from :     Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical   real-valued Wiener process defined up to time $T &gt; 0$, and let $X :  [0, T] \times \Omega \to \mathbb{R}$ be a stochastic process that is   adapted to the natural filtration $\mathcal{F}_{*}^{W}$ of the Wiener   process. Then $$      \mathbb{E} \left[ \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2} \right] = \mathbb{E} \left[ \int_{0}^{T} X_{t}^{2} \,  \mathrm{d} t \right], $$ where $\mathbb{E}$ denotes expectation with   respect to  $\gamma$. In other words, the Itō   stochastic integral, as a function, is an isometry of normed vector   spaces with respect to the norms induced by the inner products $$      ( X, Y )_{L^{2} (W)} := \mathbb{E} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) =  \int_{\Omega} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t}  \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) \, \mathrm{d} \gamma  (\omega) $$ and $$      ( A, B )_{L^{2} (\Omega)} := \mathbb{E} ( A B ) = \int_{\Omega} A(\omega) B(\omega) \, \mathrm{d} \gamma (\omega). $$   I was wondering what the two normed spaces are and what their norms are, so that the two normed spaces are isometric wrt their norms?  Thanks and regards! What is the safest way to move the iTunes Library from one hard drive to another? This may happen later with my iMac, but right now it is happening for my iTunes on PC.  The PC had 60GB of partition to install Windows 7, and right now it is down to about 2GB of free space.  All iTunes purchases, including Music and Movies, are downloaded to  C:\Users\mike\Music\iTunes\iTunes Media\Music  C:\Users\mike\Music\iTunes\iTunes Media\Movies   which will make the C: drive have less and less free space.  (and C: will be more healthy if it has at least a couple of GB of free space).  What is the safest, most proper way to migrate all the media to an external hard drive J: ?  On J:, the path might be:  J:\iTunes\iTunes Media\Music   or  J:\iTunes Media\Music   My guess is that we can change the folder in iTunes, but won't iTunes go to that folder or create it, and find nothing, and index everything wrong?  So, should an exact copy of C:\Users\mike\Music\iTunes\iTunes Media be made as J:\iTunes Media first (probably 30GB of files copying), and then start iTunes and change the preference for the folder to the J: one?  Is there a standard or official method to do this? Unintentional Lies and Misinformation Related:   I am using 1.0.71 on KitKat.  Immediately after voting, the arrow is displayed in red and the vote count is correct. However, when I scroll to read answers and then back to the top, the arrow is grey again and the vote count is reset to the previous value.  A reload of the question (pulling the finger down) is needed to show the correct situation, which is kept from then on.  This case was on security.stackexchange.com but I observed the same behavior on other sites too.  Screenshots   original state:  upvoted:  scrolled down and back up  after a reload Is there a general formula for image size vs. print size? Prove that $\sqrt{10} - \sqrt6 - \sqrt5 + \sqrt3$ is irrational I tried the methods shown in  but I cannot extend them well into 4 numbers. Mathematical induction for inequalities: $\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$ Let $f(n) = \sum_{(k,n)=1} e^\frac{2\pi ik}{n}$. Show that $f(n) = \sum_{k=1}^{n}\left(\sum_{d\mid (k,n)} \mu(d)e^{\frac{2\pi ik}{n}} \right) $ How do I disable the resizable property of a textarea? I need to refer to a model described elsewhere, from a section title, like  \section{Analysis of model X}    One way of referring to a model is to create a new counter, and then write  \section{Analysis of \cref{grandmodelofeverything}}   But now hyperref will complain because of invalid token, and it does not look good in the TOC either, especially since the titles of the sections the describe the models are more descriptive than  \section{Model 1}   They are currently like  \section{Adding the effect from Y processes}   Do I need to invent semi-descriptive long names like "The model by Authors with this modification" or can I live with these quirks? Why does unlocking the bootloader wipe phone? Particular solution of $y''-4y'+5y = 4e^{2x} (\sin x)$ How do I find the particular solution of this second order inhomogenous differential equation? (Using undetermined coefficients).  $y''-4y'+5y = 4e^{2x} (\sin x)$  I can find the generel homogenous solutions but I need help for the particular. Can one PC throw another? For example, if there is a lava river surrounding a castle. There is a drawbridge that can be lowered, but the lever is on the other side. PCs couldn't find a hidden passageway,either because PCs failed necessary check and couldn't find it or just because there isn't one. They don't have rope to try to pull the lever down. Or if PCs want to rob a house, but can't pick a lock, they see that window on the second floor is open. Can one PC use his Strength to throw another PC that is one size smaller, with other PC using his Dexterity or Strength to not fall down or lose balance? What happens after I grapple as Attack of Opportunity? CodeIgniter/PHP Active Record won't increment an integer Let $X, Y$ be independent standard normal random variables.  We already know that $X+Y$ is normal with mean 0 and variance 2. However, I am trying to prove this result in a slightly different way than usual, i.e., using convolutions.  We can address the problem in the following way: the sum of the two variables has a density $h(x)$ given by the convolution product of the densities of the two variables, say $f$ and $g$, so that   $$ h(x) = \int_{-\infty}^{\infty} f(x-y)g(y)dy. $$  However,  $$ f(y) = g(y) = \frac{1}{\sqrt {2\pi}}\exp(-y^2/2). $$  Hence now the integral becomes:  $$ h(x) =\frac{1}{2\pi}\int_{-\infty}^{\infty}\exp(-(x-y)^2/2)\exp(-y^2/2)dy. $$  However, I am puzzled here. How should we go on from here? I cannot see how to transform this integral into the integral of a normal r.v of type $N(0,2)$. Thanks in advance for your help. If I want to draw an overline over some text, not an equation, how can I do it? \overline seems to only work in math mode, and when used outside of math mode, it automatically says "Missing $ inserted". User and associated Post What is wrong with this query? (Unknown error parsing query)