How to do integration by parts when $\mu$ is a finite Borel measure. That this how to integrate by parts \begin{align} \int_0^\infty f(x) g(x) d\mu(x). \end{align}
Let $\mu$ be a general measure, suppose $f,g$ has compact support on $\mathbb{R}$, when does the integration by parts formula hold $$\int f'g d\mu = - \int g'fd\mu?$$ I know in general this is false, we can take $\mu$ to be supported on a point, say $0$, then it is not necessarily true that  $$f'(0)g(0) = -g'(0)f(0).$$  If $\mu$ is absolute continuous w.r.t. Lebesgue measure, we have $\frac{d\mu}{dx} = h$ $$\int f'gd\mu = \int f'gh dx = -\int f(gh)'dx$$ where $(gh)'dx$ might be a measure. but we can not recover the form $\int g'fh dx$.  Thank you very much!
This is similar to , but I believe it's a bit different*.  Sites like GMail support a preference that pushes all traffic through the SSL edition of the site rather than the plain-text protocol.   For sites that don't offer such preferences (or ones that may, but I have been unable to find, like facebook), is there a way using only the browser (perhaps with a plugin or addon) to always try SSL first, and fall-back to plain-text iff SSL fails?  Is that solution available on Windows, Mac OS X, and Linux? Just one?     * The previous question was looking for external applications that would accomplish this goal. $$\displaystyle\int \:\sin^n\left(x\right)\cos^m\left(x\right)\mathrm  dx=\frac{\sin^{n+1}x\cos^{m-1}x}{m+n}+\frac{m-1}{m+n}\int \:\sin^nx\cos^{m-2}x\,\mathrm dx$$  I have been trying to solve for over a week now can someone please help me. SharePoint 2013 Jan 2016 Updates Is anyone here experience problems in SharePoint after installing the latest JAN 2016 on 13th Jan 2016 (Western Australian Time) updates for SharePoint 2013. I am guessing it could be KB3114503 update giving me problems.  I am getting the same problem on other machines after running the same updates too.  The problem I am receiving is, on every list view I get this error message below:  TypeError: Unable to get property 'replace' of undefined or null reference   TypeError: Unable to get property 'replace' of undefined or null referenceSys.ArgumentNullException: Sys.ArgumentNullException: Value cannot be null. Parameter name: element CPU clock speed versus CPU core count - higher GHz, or more cores for SQL Server? We are beginning to provision a set of physical servers for a virtual cluster of SQL Server 2016 nodes within VMware. We will be utilizing Enterprise Edition licenses.  We plan on setting up 6 nodes, but there is a bit of a debate on what the ideal way to provision the physical servers with regards to CPU clock speed versus CPU core count.  I know this is largely dependent on transaction volume and number of databases stored among other software-specific factors, but is there a general rule of thumb that is advised?  For instance, is a dual 8-core, 3.2 GHz physical server (16 cores) more preferential to a dual 16-core, 2.6 GHz server (32 cores)?  Has anyone come across a white paper that further delves into this type of topic? Often (via Google Now, or when searching for a solution to some problem), I'll open a question from a non-favourite site in the app. Sometimes, I'd like to see what else is going on in the site where the question was posted. Currently, I have one of two ways:   Via tag icon:   Click one of the tag icons beneath the title Delete the [tag] text (at which point, thankfully, the sorting changes to Active from Votes).  Via manually searching for the site, which is more cumbersome. I type in hard and Hardware Recommendations is the fourth entry (below Ask Different, Raspberry Pi and Arduino)! This more annoying for meta sites, where I have to pick Meta from the dropdown menu.   Can I pretty please have a way to head to the site directly from the question? Maybe:   the site name in the top bar could become clickable and lead to the site, or an option in the More menu?     This would be especially useful for Meta sites, where questions posted around the same time could be related. Interestingly enough, most questions that Google Now are for metas of sites I visit on occasion, but haven't pinned to the list. I'm new to tikz-qtree, and can't find an automatic way to do something that qtree did very nicely.  With qtree, if you don't have a label for a single intermediate node, you get a straight line from the grandmother to the granddaughter.  With tikz-qtree, you get a "kink".  Compare the following:  With qtree:  \documentclass{article} \usepackage{qtree}  \begin{document}  \Tree [ Wiiiiiiiiiide [ nar- -row ] ]  \Tree [ nrw [ wiiiiiiiiiiiii- -iiiiiiiiiiiide ] ]  \end{document}   With tikz-qtree  \documentclass{article} \usepackage{tikz,tikz-qtree}  \begin{document}  \Tree [ Wiiiiiiiiiide [ nar- -row ] ]  \Tree [ nrw [ wiiiiiiiiiiiii- -iiiiiiiiiiiide ] ]  \end{document}   Is there any way to get the nice straight line in tikz-qtree, or do I have to adjust it manually?  Thanks in advance. How can I straighten ALL the vertical UV lines at once? Is there a hot keyshort cut? Or if I have to do them one by one .. is there at shortcut for that? Thank you! ChaCha with 64bit integers as a PRNG with streams Can I use ChaCha with four rounds as a good non-cryptographic PRNG with different streams if I use 64 bit integers instead of the standard 32 bit integers? I need a bigger state and this seems the easiest way to do it. Prove from first princibles $\frac{x^2-4}{x-4} \rightarrow -5$ as $x \rightarrow 3$ Category-theoretic limit related to topological limit? Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...?  To be more precise, is there a category-theoretic setting of some non-trivial topological space such that these different concepts of term 'limit' somehow relate?  This question came to me after I saw (  ) a surprising fact that generalised metric spaces can be seen as categories enriched over preorder $([0,\infty],\leq)$. How do you attach a debugger to a Web service? I am using a Web service. Can anybody tell me how to debug through it? I've noticed something curious about the rotation of a rectangular prism. If I take a box with height $\neq$ width $\neq$ depth and flip it into the air around different axes of rotation, some motions seem more stable than others. The 3 axes which best illustrate what I mean are:  (1) Through the centre of mass, parallel to the longest box edge.  (2) Through the centre of mass, parallel to the shortest box edge.  (3) Through the centre of mass, parallel to the remaining box edge.  It's "easy" to get the box to rotate cleanly around (1) and (2), but flipping the box around (3) usually results in extra twisting besides the rotation around (3) that I'm trying to achieve (obviously a "perfect" flip on my part would avoid this twisting, which is why I call it an instability). If you're not quite sure what I'm talking about, grab a box or book with 3 different side lengths and try it out (but careful not to break anything!).  What's special about axis (3)?   Image taken from . Similar to '', is it also ok to add an example?   I was going to  but I noticed that one answer had most of the things I needed to say, except perhaps that the questioner didn't perhaps understand the answer and  I feel that an example may help. Prove $\sum_{i=1}^n2^{i-1}=\sum_{i=0}^{n-1}2^i=2^n-1$ combinatorially.  This is easy to prove inductively.  I know that $\sum_{i=0}^n{n\choose i}=2^n$ so maybe change $\sum_{i=0}^{n-1}2^i$ to $\sum_{i=0}^{n-1}\sum_{j=0}^i{i\choose j}$.  I also know that $2^n-1$ is the number of proper subsets in a set of size $n$. How do I disable updating a form in Windows Forms? During a complicated update I might prefer to display all the changes at once. I know there is a method that allows me to do this, but what is it? Using Georeference information stored in tfwx (from ArcGIS) in QGIS? I have georeferenced some images (ArcGIS -> Update Georeferencing) which creates a *.tfwx file with the new parameters for georeferencing.   I renamed it to *.tfw but QGIS is not able to read the georeferencing information.   Does anybody has any useful tips how to fix this? I'm from the State of Kuwait and hold an Eritrean passport. I'm planning to visit UK in October but my visa will expire in November.    Is it OK to make a journey in the last month of my visa, provided that I make sure that I book my plane back before expiring date? Setting the correct encoding when piping stdout in Python Variance of Normal Order Statistics Suppose we have $X_1, \cdots, X_n \overset{\textrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$ with $n &gt; 50$, and let $X_{(1)}, \cdots, X_{(n)}$ be the associated order statistics. Are there any references pointing to formulas that specify (or estimate) the variance of the $k^{\textrm{th}}$ order statistic, i.e. $\textrm{Var}(X_{(k)})$, for $1 \leq k \leq n$? I am looking for formulas can be explicitly evaluated without resorting back to numerical integration.  I have seen this question: , which attracted a number of answers on approximate formula for the mean of the $k^{\textrm{th}}$ order statistic, e.g. Blom (1958) (and modifications based on work by Harter (1961) and Elfving (1947)). Searching on Google also yields a number of work on the variance / standard deviation for the $k^\textrm{th}$ order statistics for small $n$, all of them opting for a computation/tabulation route. They are: Godwin(1949), Teichroew (1956), and Parrish (1992). I have also attempted to derive something myself following the method sketched by @probabilityislogic in , along the lines of: $$\textrm{Var}(X_{(k)}) = \int_{-\infty}^{\infty} \left(x-\mathbb{E}(X)\right)^2 \frac{n!}{(k-1)!(n-k)!} f_X(x)[1-F_X(x)]^{n-k}[F_X(x)]^{k-1}\;\textrm{d}x,$$ with lower case $f$ denoting the PDF of a r.v. $X$, and upper case $F$ denoting the CDF. Noting $x = F^{-1}(F_X(x))$ and using the substitution $u = F_X(x)$ (with corresponding r.v. in upper case $U$), we have: $$\textrm{Var}(X_{(k)}) = \int_{0}^{1} \left(F_X^{-1}(u) - \mathbb{E}\left(F_X^{-1}(U)\right)\right)^2\,\mathcal{B}(u | k, n-k+1) \; \textrm{d}u,$$ where $\mathcal{B}$ denotes the PDF of a beta distribution (not the beta function). This can be rewritten as the variance based on the beta distribution: $$\textrm{Var}(X_{(k)}) = \textrm{Var}_{\mathcal{B}(u | k, n-k+1)}\left(F_X^{-1}(u)\right).$$ Approximating the RHS using (the variance bit of the) delta method, we get: $$\textrm{Var}_{\mathcal{B}(u | k, n-k+1)}\left(F_X^{-1}(u)\right) \approx \textrm{Var}_{\mathcal{B}(u | k, n-k+1)}(U) \cdot \left[(F_X^{-1})'\left(\mathbb{E}_{\mathcal{B}(u | k, n-k+1)}(U)\right)\right]^2,$$ where the prime denotes the derivative. The first term of the product is standard result. For the second part, given $(F^{-1})'(\cdot)=\frac{1}{f(F^{-1})(\cdot)}$ (see e.g. ), we then have: $$\textrm{Var}_{\mathcal{B}(u | k, n-k+1)}\left(F_X^{-1}(u)\right) \approx \frac{k(n-k+1)}{(n+1)^2(n+2)} \frac{1}{\left(f\left(F^{-1}\left(\frac{k}{n+1}\right)\right)\right)^2}.$$ Substituting in the standard normal distribution ($\phi$ for PDF, $\Phi$ for CDF) and scaling it to the desired variance $\sigma^2$ we arrive at: $$\textrm{Var}(X_{(k)}) \approx \frac{k(n-k+1)}{(n+1)^2(n+2)} \frac{1}{\left(\phi\left(\Phi^{-1}\left(\frac{k}{n+1}\right)\right)\right)^2},$$ which we can throw in a $\sigma^2$ term in front if it is not a standard normal. But surely someone must have derived that somewhere (again I am looking for some references), and there is a high chance that I have gone off track somewhere above. I will be getting a point-and-shoot camera for taking portraits and only portraits.  Specifically portraits with a shallow depth of field.   Groups portraits sometimes.  It's just too much work to add lens blur at post-prod.  I understand that setting a P&amp;S on macro mode gives a shallow focus and DOF.    How do I read P&amp;S camera specs to shortlist those with the shallowest depth of field?   Added: By point-and-shoot, I mean a camera that is easy to use -- Not necessarily compact, inexpensive, or current.  Also, if there are two or three factors/specs to get a shallow depth of field, then which spec contributes the most to bokeh? which is second most contributing? the third, etc.?