I am trying to evalute this integral: $$ \int_{A} x^{m} y^{n} z^{p}(1-x-y-z)^{q}|d x \wedge d y \wedge d z| $$ A=R3 , x , y and z are positive and x+y+z&lt;=1 I used the change of values u=x+y+z , v=(z+y)/u and $\omega$=z/(y+z) which gave me x=u(1-v) y=vu(1-$\omega$) and z=uv$\omega$ The det of the matrix of the change of variables is: u^2*v However, I am struggling to find the boundries for u, v and $\omega$ and carrying the calculation. Any help would be appreciated.
Let $A =$ {$(x,y,z) \in \mathbb{R}^3 |$  $x,y,z \geq 0$ and $x+y+z \leq 1$}.   Compute the integral $$\int_{A} x^m y^n z^p (1-x-y-z)^q dx dy dz$$  using new variables $u = x+y+z$, $v=\frac{y+z}{x+y+z}$, $w=\frac{z}{y+z}$.  Also, although it was not stated in the problem, I'm assuming that $m,n,p,q \in \mathbb{Z}$.   Since $f(x,y,z)=(x+y+z, \frac{y+z}{x+y+z}, \frac{z}{y+z}) = (u,v,w)$, $f^{-1}(u,v,w) = (u(1-v), uv(1-w), uvw)$  Thus J $= \begin{vmatrix} 1-v &amp; -u &amp; 0\\  v(1-w) &amp; u(1-w) &amp; -uv\\  vw &amp; uw &amp; uv\\ \end{vmatrix} = u^2v$  Skipping the part where I find the new limits of integration, and skipping some algebraic simplification we have:  $$\int_{A} x^m y^n z^p (1-x-y-z)^q dx dy dz$$   $$= \int_{0}^1 \int_{0}^1 \int_{0}^1 u^{m+n+p+2}(1-u)^q v^{n+p+1}(1-v)^mw^p(1-w)^n du dv dw$$  However, I have no  idea where to go from here. Any help would be much appreciated.
Autoexpand UITextView height dynamically in ios Was Tracy Mills a part of John Doe's plan from the beginning? In , John Doe is shown as being extremely meticulous and patient while carrying out his plan of murdering and/or torturing people in accordance with the seven deadly sins.  At the end of the film, it is revealed that John Doe had killed and decapitated Tracy Mills, the wife of Detective David Mills.  Was Tracy Mills his intended target from the beginning?  Did John Doe decide to kill Tracy only after Detective Mills and Somerset discovered his residence? How do you determine what technology a website is built on? Quite often I come across a nice looking or functional website, and wonder what technology was used to create it. What techniques are available to figure out what a particular website was built with?  Few frameworks seem to include any kind of generator meta tag like web editors do. Are there any tell-tale signs of particular languages and/or frameworks?    Summary of answers  Site URLs may betray the framework and/or programming language but cannot be relied upon (e.g. file extensions such as .asp). HTTP response headers, cookies, stylesheets and source comments may also give clues.  Some nice tools for querying site details (no doubt there are many more):          Firefox addons:    - CMS, frameworks/libraries, e-commerce, message boards etc.  - IP, country and webserver details  - Javascript libraries in use   Chrome Extensions:        Bookmarklets:    - shows Javascript framework in use Ubuntu 16.04 hangs on splash screen during boot In my case I'm talking about Assassin's Stance, but I'd like to know the answer for stances in general.  I feel like I might be missing something incredibly obvious caused by missing a word or two (or a paragraph), but I can't find anything that explicitly states whether or not I can use the Tome of Battle stances with ranged weapons like the bow (or crossbow).  I realize that half the classes have "blade" in their names so using ranged weapons is not in the spirit of things, but I'm looking for an explicit answer (or as close to one as possible) from any WotC official resource. If this has already been answered I'm sorry, but I could not find the answer myself from searching. Some GUI apps use OS X location services, but I want to retrieve a Mac's physical location from the . This could be useful for running scripts, switching settings etc. The mechanism should be , not a Geo IP service (data too low-res, needs online connection).  Minimum requirement is output of longitude and latitude. Nice would be a number indicating the precision. Super nice: a geocoder that provides location names. How to recover data from a failing hard drive? An external 3½" HDD seems to be in danger of failing — it's making .  I've acquired a replacement drive, and want to know the best strategy to get the data off of the dubious drive with the best chance of saving as much as possible.  There are some directories that are more important than others.  However, I'm guessing that picking and choosing directories is going to reduce my chances of saving the whole thing.  I would also have to mount it, dump a file listing, and then unmount it in order to be able to effectively prioritize directories.  Adding in the fact that it's time-consuming to do this, I'm leaning away from this approach.  I've considered just using dd, but I'm not sure how it would handle read errors or other problems that might prevent only certain parts of the data from being rescued, or which could be overcome with some retries, but not so many that they endanger other parts of the drive from being saved.  I guess ideally it would do a single pass to get as much as possible and then go back to retry anything that was missed due to errors.  Is it possible that copying more slowly — e.g. pausing every x MB/GB — would be better than just running the operation full tilt, for example to avoid any overheating issues?  For the "where is your backup" crowd: this actually is my backup drive, but it also contains some non-critical and bulky stuff, like music, that aren't backups, i.e. aren't backed up.  The drive has not exhibited any clear signs of failure other than this somewhat ominous sound.  I did have to fsck a few errors recently — orphaned inodes, incorrect free blocks/inodes counts, inode bitmap differences, zero dtime on deleted inodes; about 20 errors in all.  The filesystem of the partition is ext3. What should I do if I want to be able to run a given program regardless of my current directory? Should I create a symbolic link to the program in the /bin folder? What happens if you use the 32-bit int 0x80 Linux ABI in 64-bit code? Intersection of powers of an ideal in a Noetherian ring Given a Noetherian ring $R$ and a proper ideal $I$ of it.  Is it true that $$\bigcap_{n\ge 1} I^n=0$$ as $n$ varies over all natural numbers? If not, is it true if $I$ is a maximal ideal? If not, is it true if $I$ is the maximal ideal of a local ring $R$? If not, is it true under additional assumptions on $R$ (like $R$ is regular)? Looking for qt-at-spi for ubuntu 20.04 I was able to read and write NTFS filesystems normally in Mountain Lion, but after the upgrade this stopped to work.  I installed ntfs-3g and fuse4x using homebrew in Mountain Lion and followed the instructions displayed by homebrew to finish the installation of fuse4x kernel extension.  To try to solve this problem I removed ntfs-3g, fuse4x and fuse4x-kext and reinstalled them. But this didn't solve my problem.  Does anyone know how to solve this? Thanks. In transformers, the ratio of the voltages equals the ratio of the turns - so double the output coil's turns and the output voltage doubles. Then, in order to conserve energy, current halves.  This makes perfect sense in terms of $\mathrm{P=VI}$, but what happened to $\mathrm{V=IR}$? Doubling voltage and halving the current seems to completely contradict this basic law. That is, of course, unless the resistance in the output circuit changes, with R proportional to $\mathrm{V^2}$ - but I don't see how this is possible.  So how can a transformer obey both laws? Can resistance change or am I missing something else? Show that a norm on an inner product space satisfies parallelogram law. Hence use the parallelogram law to show that the space of continuous real functions defined on the interval $[a,b]$ is not a Hilbert space. Here I did the first part. Let $X$ be an inner product space and $x,y\in X$ consider $\Vert x+y\Vert ^2$ and $\Vert x-y\Vert ^2$ Then on adding I get $\Vert x+y\Vert ^2+\Vert x-y\Vert ^2=2(\Vert x\Vert ^2 +\Vert y\Vert ^2)$ For the second part I don’t know how can a parallelogram law prove $C[a,b]$ is not a Hilbert space, help me please. What software can I use to write on PDFs? I'm used to OS X where the system PDF viewer will let you draw and write text on top of any PDF document. You can then flatten it and save as a new PDF. It saves paper and time so you can fill in any form without printing it and scanning it back to the computer.  It also lets you insert a scanned image of your signature.  Is there software for Ubuntu that would do that? Simpler proof - Non atomic measures Suppose that $(X,\mathcal{E},\mu)$ is a non-atomic finite measure space (i.e. for every $E \in \mathcal{E}$ with $\mu(E)&gt;0$ there exists $F \subset E$ measurable such that $0&lt;\mu(F) &lt;\mu(E)$.)  a) Prove that for every $ \varepsilon &gt;0$ there is a finite partition of $X$ in measurable subsets $X_1,..,X_n$ such that $\mu(X_i)\leq \varepsilon$.  b) Prove that for every $\alpha \in [0,\mu(X)]$ there exists $E \in \mathcal{E}$ with $\mu(E)=\alpha$.    I guess that a) is given to prove b) more easily. I have the following idea of solution for a) (inspired from , which proves b)  )  Denote  $$ \Gamma = \{ (X_1,..,X_n) : n &gt;&gt; \mu(X)/\varepsilon, X_i \text{ are disjoint }, \mu(X_i)\leq \varepsilon \}$$ ordered by componentwise inclusion. Totally ordered parts $(Y_\alpha)$ of $\Gamma$ have an upper bound element $(\bigcup Y_\alpha^i)_{i=1}^n$ which is still in $\Gamma$.   By Zorn's lemma $\Gamma$ has maximal elements. If a maximal $(X_1..X_n)$ element is not a partition, then we can replace it with something better, of the form $(X_1,..,X_n\cup A)$ where $A \subset X \setminus (X_1 \cup..\cup X_n)$ and $\mu(A)&gt;0$.     My questions are:  1) Is my solution of a) correct? I feel that the part with the upper bound element may not work, since there might be noncountable unions.  2) What is a simpler solution of a)? Does a gluon have mass or not? I was under the impression that gluons had no mass. Then Wikipedia throws this curveball:  Mass: 0 (theoretical value) &lt; 0.0002 eV/c2 (experimental limit)  Which one should a poor soul go by? If the theoretical value just the one that people &quot;thought&quot; was right, until they found out the truth, or is the experimental limit(?) an impossibility that still has not been explained? Or something third? An elementary question regarding a multiplicative character over finite fields Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I have no background in Algebraic Number Theory, but if someone explains in that language, I would like to see it and learn it as much as I can at this moment. References will also be appreciated!  Let $q$ be a power of prime $p$ such that $q \equiv 1 \mod 4$ and $p \nmid 2n$.  Question. Let $\chi_{4} : \mathbb{F}_{q}^{*} \rightarrow S^{1}$ be the character of order $4$, show that $\chi_{4}(-4) = 1$.  The hint in the book says we may show that $\chi_{4}(4) = \chi_{4}(-1) = 1$ for $q \equiv 1 \mod 8$ and $\chi_{4}(4) = \chi_{4}(-1) = -1$ for $q \equiv 5 \mod 8$.  If $q \equiv 1 \mod 8$, then $\chi_{4}(4) = \chi_{4}(2)^{2} = \chi_{2}(2) = \pm 1$, where $\chi_{j}$ denotes the character of order $j$. So I think $2$ must be quadratic residue in this case, but I cannot figure out why. I am not mathematician by any means so this question might be rather stupid. I came across this Wikipedia article on  and found this bewildering formula,   $$1 + 2 + 3 + \dots  = - \frac1{12}$$  The article also says that "Ramanujan summation of a divergent series is not a sum in the traditional sense". I am wondering then what this summation actually implies?  Is there any interpretation of this confounding result in the physical world that someone not so math-savvy can understand? How to convert an instance of std::string to lower case I want to convert a std::string to lowercase. I am aware of the function tolower(). However, in the past I have had issues with this function and it is hardly ideal anyway as using it with a std::string would require iterating over each character. Is there an alternative which works 100% of the time?