Update OpenGL on Intel Integrated Graphics?
How can I update my OpenGL to 3.0 or higher
How to design a product table for many kinds of product where each product has many parameters Multiple Image in Canvas ItemsControl I recently saw a lecturer prove the following theorem (assuming the result that every analytic function is locally 1-1 whenever its derivative is nonzero): Let $\Omega \subset \mathbb{C}$ be open, and let $f : \Omega \to \mathbb{C}$ be 1-1 and analytic on $\Omega$.  Then $f&#39;(z_0) \not = 0$ for every $z_0 \in \Omega$.  I got the basic idea behind the proof: we assume for contradiction that $f&#39;(z_0) = 0$, and, assuming without loss of generality that $z_0 = f(z_0) =0$, we have (from the power-series expansion) that $f(z) = z^kg(z)$ for some analytic $g$ in some disk at the origin (i.e., $z_0$) and some $k \ge 2$. Since $z^k$ is not 1-1 in any such disk (because there are multiple roots of unity), then $f$ isn't either.  However, the proof he gave was rather awkward and technical- it involved defining three different axillary functions, even though the idea was simple, and I've since forgotten how it exactly worked.  In any case, I'm convinced there's a better way.  The problem is that I'm having trouble turning the idea into a real proof- I know that it obviously follows if $g$ is 1-1, but I'm also pretty sure that that is too strong an assumption.  Am I missing something, or does the argument just have to be more complicated? How is the nature of logical principles commonly defined in contemporary philosophy? In contemporary philosophy, how exactly is the nature of logical principles defined? For example, the way I've commonly seen logical principles construed are as true propositions which described the most general laws of reality. The previous definition does seem very intuitive to me; if we understand logical principles as being concerned with truth, and if we accept a correspondence theory of truth, then logical principles simply describe fundamental laws regarding various states of affairs occurring in reality. (e.g. If state of affairs S1 obtains and state of affairs S2 obtains, then state of affairs S3 must also obtain given logical law L).  However, I've mostly heard the previous view expressed among laypersons. I'm interested to know how most philosophers would think about the nature of logical principles. I finally got dragonrend and defeated Alduin, but I'm still too weak for fighting ancient dragons. Any certain weapon for killing them.  (I've got lvl 20 one handed weapons and lvl 46 for archery, so the Dragonbane is out of the question) How to set render engine in blender? Can a single molecule have a temperature? Have we managed to make a perfect vacuum? Have we managed to make a device without any atom inside it on earth? I was reading about vacuum , and I found in the examples part  that even on the best man made vacuum devices, there are still millions of "molecules per $cm^3$". So my other question here is, what are these molecules and what is the mass of them? Proving a distance between molecules defines a metric space. A DNA molecule can be represented as a string of symbols $A$, $C$, $G$ and $T$, such as  $$GGATAATTCTAG. . .GACCGTACCC$$ For the purposes of this question, we will assume that all DNA molecules contain the same (large!) number $N$ of symbols. Thus, a DNA molecule is an $N$-tuple $x = (x_1, . . . , x_N)$ where $x_i  ∈ \{A,C,G,T\}$ for each $i$.  Define the distance between two DNA molecules $x$, $y$ as the number of elements $i ∈ {1, . . . ,N}$ such that $x_i \neq y_i$. Prove that this defines a metric on the set of DNA molecules. Finding the id of a parent div using Jquery My company has purchased a product that renders an ASP.NET control on the page. This control uses jQuery 1.2.3 and adds a script tag to the page to reference it. The developers of the control will not support use of the control if it modified in any way (including modification to reference a different version of jQuery).  I'm about to start development of my own control and would like to use the features and speed improvements of jQuery 1.3. Both of these controls will need to exist on the same page.  How can I allow the purchased control to use jQuery 1.2.3 and new custom development to use jQuery 1.3? Also out of curiosity, what if we were to use an additional control that needed to reference yet another version of jQuery? The final state of 1000 light bulbs switched on/off by 1000 people passing by How do you make HDDs and partitions mount at startup on Ubuntu?  I always keep my media and data files on separate partitions - one 2tb HDD and a 400 odd gig partition.  I have been trying to swap the default folder location for the home directory to my other drives.  It works when I change the /home/user_name/.config/user-dirs.dirs file to the directories I want, but only until I reboot the system. I am only assuming that the drives not mounting is the problem with it, but it would be helpful either way.  If any one knows another reason for why the directories change back to default each time that would be good also. I know that $$\gcd(a^b-1,a^c-1)=a^{\gcd(b,c)}-1$$ which can be seen by expanding both terms into geometric series. Is there any such simplification for $$\gcd(a^b+1,a^c+1)=?$$  EDIT: When working with polynomials instead of whole numbers, it is $$gcd(x^n+1,x^m+1)=\begin{cases}a^{\gcd(n,m)}+1 &amp; \nu_2(n) =\nu_2(m)\\1 &amp;\text{else}\end{cases}$$ But this gives only a lower bound for the whole-number case. I had a bunch of errors in the output of sudo apt-get update:  duplicate entries and 404 not found errors.  Figured those out, I hope: I removed PPAs that didn't have any packages associated with them, then I removed out a duplicate entry in /etc/apt/sources.list.  It seems I have royally buggered it up.  Now most of the lines say Ign when I do sudo apt-get update.  How do I go about straightening out what I have screwed up?  I need to get rid of these Igns that are all over the place.  There are about 50% (eyeball guess) Igns! What does an ampersand after this assignment operator mean? How to prevent unification when animating in photoshop? I am trying to create an animated gif where text drops down and smashes into the background and cracks it. Whenever I try and transform it though, the transformation is applied to all frames. How do I prevent this? I'm using Adobe Photoshop CS6 (64-bit) Transit visa at Istanbul Airport, Turkey, during a Turkish Airlines arranged stopover? Do I need a transit visa at Istanbul Airport to enter Istanbul for city sightseeing or hotel accommodation?. All services provided free by the Turkish Airline. I am Nigerian citizen and booked Turkish Airline flight to Nigeria with 12hours layover at Istanbul Airport. Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$ Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number.  The $A(p,q)$'s are known as alternating .     Can someone provide a nice proof that    $$A(1,1) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2?$$   I worked for a while on this today but was unsuccessful.  Summation by parts, swapping the order of summation, and approximating $H_k$ by $\log k$ were my best ideas, but I could not get any of them to work.  (Perhaps someone else can?)  I would like a nice proof in order to complete .  Bonus points for proving $A(1,2) = \frac{5}{8} \zeta(3)$ and $A(2,1) =  \zeta(3) - \frac{1}{2}\zeta(2) \log 2$, as those are the other two alternating Euler sums needed to complete my answer.   Added: I'm going to change the accepted answer to robjohn's $A(1,1)$ calculation as a proxy for the three answers he gave here.  Notwithstanding the other great answers (especially the currently most-upvoted one, the one I first accepted), robjohn's approach is the one I was originally trying.  I am pleased to see that it can be used to do the $A(1,1)$, $A(1,2)$, and $A(2,1)$ derivations. What library do I have to use such that the document can render \lt and \gt as less than and greater than signs, respectively? \documentclass{article} %\usepackage{amsmath} It does Not work ! \begin{document} \[ x\lt y \gt z \] \end{document}  What library do I have to use such that the document can render \lt and \gt as less than and greater than signs, respectively? Edit I am using MathJax library in my site under Asp.net Mvc 3. Web application is vulnerable to XSS and XSRF attacks so &lt; and &gt; should be avoided as the input from the users. My database will save LaTeX commands that can be rendered by both LaTeX compiler and MathJax. Therefore, I need \gt and \lt that already work in MathJax. Source:  Rendered: