Artifacts in rendered geometry
Cause of black artifacts in render?
How can I take a picture with my webcam in Windows 7? I know it existed as sort of a folder in Windows Explorer in at least Windows XP, but I just can't seem to find it in Windows 7...  How can I take a picture with my webcam in Windows 7 without using any 3rd party tools? My question concerns the inductive step of the proof below. I've typed out the work I've done so far. If anyone has any suggestions on how to proceed thanks in advance.  If $n\in\mathbb{N}$, then $(1-\frac{1}{2})(1-\frac{1}{4})(1-\frac{1}{8})(1-\frac{1}{16})\cdots(1-\frac{1}{2^n})\ge\frac{1}{4}+\frac{1}{2^{n+1}}$.  $Proof.$ We use mathematical induction.  Base Case. For $n=1$ we see that $1-\frac{1}{2}\ge\frac{1}{4}+\frac{1}{4}$ since $\frac{1}{2}\ge\frac{1}{2}$.  Inductive Step. Let $n&gt;1$. Assume our statement is true for $n$. Now observe that  $\begin{align}(1-\frac{1}{2})(1-\frac{1}{4})(1-\frac{1}{8})(1-\frac{1}{16})\cdots(1-\frac{1}{2^{n+1}})&amp;\ge\\ \left((1-\frac{1}{2})(1-\frac{1}{4})(1-\frac{1}{8})(1-\frac{1}{16})\cdots(1-\frac{1}{2^{n}})\right)(1-\frac{1}{2^{n+1}})&amp;\ge(\frac{1}{4}+\frac{1}{2^{n+1}})(1-\frac{1}{2^{n+1}}) \end{align}$  My question is: How do we get from $\begin{align} (\frac{1}{4}+\frac{1}{2^{n+1}})(1-\frac{1}{2^{n+1}}) \end{align}$ to $\frac{1}{4}+\frac{1}{2^{(n+1)+1}}$?  Most of what I've attempted has led to tedious calculations that do not simplify to the desired result. I've also tried making the argument that $(1-\frac{1}{2^{n+1}})$ must be $\ge$ to $\frac{1}{2}$ but then I'm stuck with $\frac{1}{8}+\frac{1}{2^{(n+1)+1}}$... Is my approach incorrect here? Calculate integral $\int_{-1}^{1} (1-x^2)^n dx $ Declare global variables in Visual Studio 2010 and VB.NET "...would have trouble imagining...": Is this present participle or gerund? sudo: /var/lib/sudo writable by non-owner (040777), should be mode 0700 I get error message always when I work with sudo command.  It shows always sudo: /var/lib/sudo writable by non-owner (040777), should be mode 0700.   A while ago I changed the permission of /var folder to 777.   How can I fix it? Illustrator: how to change the artboard number? mongodb group values by multiple fields This is all in iTerm2 on OS X.  I used to have  export TERM='xterm-256color'   in my .bashrc. This meant Vim in tmux did use 256 colors.  And once I added  set -g xterm-keys on   then keyboard shortcuts with modifiers worked fine in Vim. Namely: shift+left/right that I mapped to switch Vim tabs, ctrl+up/down that I mapped to move ("bubble") lines, and shift+left/right which worked out of the box to jump by word in the command-line mode (e.g. when typing something like :e foo bar baz).  However, this setup had the problem that the Vim background color only shows behind text, as mentioned .  So I removed  export TERM='xterm-256color'   from my .bashrc and instead put this in my .tmux.conf:  set -g default-terminal "screen-256color"   That fixed the Vim background color, but broke the keyboard shortcuts - they do unexpected things (move the cursor, delete text) instead.  By using ctrl+v to insert the verbatim output from the key combinations (as described ), I was able to work around it:  map [1;5A &lt;C-Up&gt; map [1;5B &lt;C-Down&gt; map [1;2D &lt;S-Left&gt; map [1;2C &lt;S-Right&gt; cmap [1;2D &lt;S-Left&gt; cmap [1;2C &lt;S-Right&gt;   This makes the shortcuts work, but it doesn't feel like the right solution. Could anyone tell me what's happening here and how to fix it? What makes matter travel along geodesics? The relativistic explanation of gravity is geometric, the motion of a body in a field of space-time distortion can be described as being at rest and travelling along a geodesic of that field, but why does it travel along that geodesic? What causes it to move? How to prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$? How do I prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$? for $x, y$ positive?  This should be easy, but I'm not seeing how.  A hint would be appreciated. Are fractions with prime numerator and denominator dense? So the question is in the title. And I mean dense in positive real numbers ofcourse. Somehow I cannot grasp if this is very trivial or not. The prime numbers aren't that dense, but are there enough of primes to make the fractions constructed from them dense? What is the difference between class and instance attributes? I have Dubai work permit mailed to me by my employer. It contains my old passport number which has ECR stamp. I then got a new passport issued without ECR stamping, but with different passport number. I have both passports.  Can I travel to Dubai now? Why does my user number show up in the search box when clicking my profile information? If you could travel to the center of the Earth (or any planet), would you be weightless there? ref to subsection number only A word similar to "Donate" but more freely provided to How to fix TikZ corners in 3D Am trying to draw a simple rectangle and there seems to be minor issue with two of the endpoints (f and d in the following). I tried line cap=rect as suggested by  but that did not seem to work.  The resulting image:    \documentclass{standalone} \setlength\PreviewBorder{5pt}%  \usepackage{tikz}  \newcommand*{\Width}{2.0} \newcommand*{\Height}{3.0} \newcommand*{\Depth}{1.0}  \begin{document} \begin{tikzpicture}%     \coordinate (A) at ( 0,       0,     \Depth);%     \coordinate (B) at (\Height,  0,     \Depth);%     \coordinate (C) at (\Height,  0,      0);%     \coordinate (D) at (\Height, \Width,  0);%     \coordinate (E) at ( 0,      \Width,  0);%     \coordinate (F) at ( 0,      \Width, \Depth);%     \coordinate (G) at (\Height, \Width, \Depth);%     \coordinate (H) at ( 0,       0,      0);%     %     \draw [blue, ultra thick,line cap=rect]%         (A) -- (B) -- (C) -- (D) -- (E) -- (F) -- cycle;%     \draw [blue, ultra thick,line cap=rect]%         (B) -- (G) -- (F) -- (E) -- (D) -- (G) -- cycle;%      \path (D)  [right] node {$d$};% Problem Point 1     \path (F)  [left ] node {$f$};% Problem Point 2 \end{tikzpicture}% \end{document}   This problem about  seems related, but I do not understand how to adapt that to my problem. Example of a group where $o(a)$ and $o(b)$ are finite but $o(ab)$ is infinite Let G be a group and $a,b \in G$. It is given that $o(a)$ and $o(b)$ are finite. Can you give an example of a group where $o(ab)$ is infinite?