Source of Light bending
How is light affected by gravity?
What original Dr Who episodes are left when not watching unfinished stories? Integer coefficient polynomial $p(x)$ has no integer roots of $\,p(0)$ and $p(1)$ are odd [Parity Root Test] This came up an a training piece for the Putnam Competition and also in Ireland and Rosen.  The question posed was basically:  Let $p(x)$ be a polynomial with integer coefficients satisfying that $p(0)$ and $p(1)$ are odd. Show that $p$ has no integer zeros.  I&amp;R give an example:  $p(x) = x^2 - 117x + 31$  and show (no problem) that for any $n$ whether even or odd, $p(n)$ will be odd. And claim that this shows $p(n)$ will never be $0$.  I can see, e.g., that $x^2 + 2x + 1$ will be odd substituting an even $n$ and even for an odd $n$.   But would appreciate help in understanding the underlying math and what is happening here.  Also, as a second part, can a general statement about the existence of an integer solution be made if $n$, even and odd, generates an even and an odd as in the last example.  I can see that if you look at these equations (mod $2$), you can distinguish whether there is an integer solution. And I would guess this is intimately connected with the question.  Thanks as always. I have a C++ library that provides various classes for managing data.  I have the source code for the library.  I want to extend the C++ API to support C function calls so that the library can be used with C code and C++ code at the same time.  I'm using GNU tool chain (gcc, glibc, etc), so language and architecture support are not an issue.  Are there any reasons why this is technically not possible?  Are there any gotcha's that I need to watch out for?  Are there resources, example code and/or documentation available regarding this?    Some other things that I have found out:   Use the following to wrap your C++ headers that need to be used by C code.     #ifdef __cplusplus extern "C" {   #endif   //   // Code goes here ...   //   #ifdef __cplusplus   } // extern "C"   #endif    Keep "real" C++ interfaces in separate header files that are not included by C.  Think  here.  Using #ifndef __cplusplus #error stuff helps here to detect any craziness. Careful of C++ identifiers as names in C code Enums varying in size between C and C++ compilers.  Probably not an issue if you're using GNU tool chain, but still, be careful. For structs follow the following form so that C does not get confused.  typedef struct X { ... } X  Then use pointers for passing around C++ objects, they just have to be declared in C as struct X where X is the C++ object.   All of this is courtesy of a friend who's a wizard at C++. Unary operator expected #!/bin/bash  SUBJECT="WARNING CPU USAGE HIGH" TO=gmail id MESSAGE=/tmp/messages echo "#######################" &gt; $MESSAGE echo "CPU statistics as follows.." &gt;&gt; $MESSAGE mpstat &gt;&gt; $MESSAGE echo "#######################" &gt;&gt; $MESSAGE CPU_USAGE=$(top -b -n1 | awk '/^Cpu/ {print $2}' | cut -d. -f1) [ $CPU_USAGE -gt 85 ] &amp;&amp; mail -s "$SUBJECT" "$TO" &lt; $MESSAGE`   ./cpu.sh: line 11: [: -gt: unary operator expected What might be the reason I have a list of structs and I want to change one element. For example :  MyList.Add(new MyStruct("john"); MyList.Add(new MyStruct("peter");   Now I want to change one element:  MyList[1].Name = "bob"   However, whenever I try and do this I get the following error:     Cannot modify the return value of   System.Collections.Generic.List.this[int]‘ because it is not   a variable   If I use a list of classes, the problem doesn't occur.  I guess the answer has to do with structs being a value type.  So, if I have a list of structs should I treat them as read-only? If I need to change elements in a list then I should use classes and not structs? I saw this movie a while back. I don't remember how it starts, but from where I remember watching it, it may have not been too far from the start. Apparently, Earth has been conquered by some alien empire or something and humanity are basically cavemen being used as slaves. From what I remember, the main antagonist alien is seen taking one of the slaves to a room where a machine is used on him to make him smart; this is apparently a big no-no so the main antagonist and his sidekick have to keep it quiet. The idea was that by making one of the slaves smart, it could increase the work output of the others. Apparently the antagonist is using the slaves on Earth to gather gold bullion  which is apparently still very valuable to the aliens but he's doing it illegally (not the slave labor; that's apparently ok). The protagonist who had the machine used on him becomes the intermediary between the slaves and the aliens and begins teaching the other slaves. I remember him teaching them mathematics and I remember him saying that it was universally accepted (I think he may have been referring to algebra or trigonometry). At some point he becomes aware that they are slaves and organizes a plan to free not only his fellow slaves but the Earth. Some escape and find an old museum which has old flight simulators, and shows that humanity wasn't always a slave race and that it may have been modern times in which Earth was invaded (i.e. no lasers or starships). I think they also find Fort Knox as there is a huge collection of gold bricks and weapons. The protagonist comes up with a plan to destroy some sort of teleporter the aliens use to transport stuff to and from Earth, however that alone wont be enough as they could fight back, which will be bad for humanity, which is still undeveloped. So the plan also includes using the teleporter before destroying it to to transport and detonate a nuclear warhead on the alien home world which apparently cripples any attempt to immediately send a fleet but also as a show of strength. After executing their plan, the main antagonist is captured by the humans and as an insult is locked in a cage surrounded by the gold he sought. If I recall correctly the place is set to detonate as to destroy the precocious gold and that will be used against the aliens should they mobilize a fleet in the future (i.e. attack us and we destroy Earth's supply of gold). The antagonist's side kick for some reason I remember as a big black man from whom the antagonist had removed his hand at some point and also apparently the antagonist was cruel to him. After the antagonist is locked up, he is trying to appeal to his sidekick's loyalty, however the sidekick takes the humans' side, I'm also getting an image that apart from treating him kindly the sidekick's missing hand was restored. Solutions to $a+b+c=12$, $a, b, c \in \mathbb{N}_0$ Let $a, b, c \in \mathbb{N}_0$. If $a+b+c=12$, how many solutions $(a,b,c)$ satisfy the equation?   Is the answer:      $729$ How to add shortcut keys for java code in eclipse More options when flagging for migration I think there should be more Stack Exchange networks available when flagging a question to be migrated. For example I'd think that the following question:    ...would be a better fit on unix.stackexchange.com instead of Stack Overflow. But without the option to mark it so, I feel hesitant to mark it. Creating Network Flow Diagram using QGIS I have a Telecom cable network in Geopackage format. How do I create a Network flow diagram from which to represent the relationship between Fiber distribution hub to Network access point? You can use the range feature of unicode-math to set the math font for a specific unicode slots or the macro accessing the glyph. I tried this with the sum sign &amp; Latin Modern, see my example:  \documentclass{scrartcl}  \usepackage{amsmath} \usepackage{unicode-math} % use unicode in math &amp; setup math fonts \setmathfont{Asana Math} %\setmathfont[range=\sum]{Latin Modern Math} % substitute Sum sign  \begin{document}  Notice the different sum sign $\sum$, as intended.   Notice also, the difference in spacing, not intended: \begin{equation}     \begin{bmatrix}         0           &amp; \frac{1}{2}   &amp; \frac{2}{3} \\         \frac{3}{4} &amp; 0             &amp; \frac{4}{5} \\         \frac{5}{6} &amp; \frac{6}{7}   &amp; 0     \end{bmatrix}     \begin{pmatrix}         A \\         B \\         C     \end{pmatrix} \end{equation} \end{document}   However, not only the sum sign, but also the spacing in the matrices changes. With Latin Modern commented out (as in the example):  With Latin Modern active (remove % before relevant line):  Why is that? BTW, I'm using Lualatex from TeXLive 2012. This is frustrating for me. The UV-textures to vertex cols add-on doesn't seem to work despite the specific instructions. Any other tutorials I can find are either very old or very poor.  I'd like to know how to do this in Blender 2.79. I need vertex colors in my ambient occlusion for use in Shader Forge. How can I edit/create new launcher items in Unity by hand? Will Unity allow making custom launcher icons from .desktop files or via menu editing system? (Right now the launcher doesn't give the option to "keep in launcher" on all programs.  For some programs I use, I have to make custom launchers or .desktop files.  For instance, daily blender builds are generally just folders with an executable.  In basic Gnome or KDE, I can make a new menu entry with the menu editing system. Then, I can also add it to Docky either from the menu or by dragging a .desktop file to it. Unity launcher doesn't support drag and drop, so thats not a bug or anything, but when I open a .desktop file, it has unpredictable results. Most time it will not have "keep in launcher". Sometime it will have a pinnable item without the .desktop's icon, and if I pin the item to the launcher, it will not call upon the program again after closing it. I've also gotten it to just work with a .desktop file for celtx. Sorry for the title, it is not very evident and intuitive but I really do not how to tell it better...  Well, you know, several times, or better, many times, we use this form:  If I want to say:      "development of special weapons was   the first point in Hitler's   program..."   I will say this (a better form):      "special weapon development was..."   OK...  is it      "special weapon development"   or      "special weapons development" (note   the plural...)   what's the grammar rule in order to understand how to use this very used form? Coloring lines inside textarea Raise bounty on question I have this question here:  I chose 500 then before accepting decided that was too much and halfed it, but now I just want the answer so I'd like to raise it to 500 but I can't find how to it. Also it's just such an irritating problem that I'd give all my rep away if I could to the clever person who has the answer. It would be nice to have a few minutes after creating the bounty to be able to raise it. But for now could a mod or something please raise it for me to 500. Prove $\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots$ converges to $\frac 1 2 $ Show that $$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots = \frac{1}{2}.$$  I'm not exactly sure what to do here, it seems awfully similar to Zeno's paradox. If the series continues infinitely then each term is just going to get smaller and smaller.  Is this an example where I should be making a Riemann sum and then taking the limit which would end up being $1/2$? See how many duplicates of a post? Can I see how many duplicates there are of a post? I.e. how many posts have been labelled as duplicate because of this post? If so, how? And can I also get a list of these duplicated posts? Sort and count number of occurrence of lines I have Apache logfile, access.log, how to count number of line occurrence in that file? for example the result of cut -f 7 -d ' ' | cut -d '?' -f 1 | tr '[:upper:]' '[:lower:]' is  a.php b.php a.php c.php d.php b.php a.php   the result that I want is:  3 a.php 2 b.php 1 d.php # order doesn't matter 1 c.php Is an ideal which is maximal with respect to the property that it consists of zero divisors necessarily prime? This is in follow-up to .     Let $R$ be a commutative ring with identity and consider the set $Z \subset R$ of zero divisors. If the ideal $I\subset Z$ is maximal with respect to the constraint, need it be prime?    Since Zorn's Lemma applies equally well to proving that either minimal prime ideals exist or that maximal ideals contained in $Z$ exist, this would provide an alternative proof of that question if true.  A naive approach to proving this is to assume $ab \in I$ with $a, b \notin I$. Since $ab \in I$, it follows that $ab$ is a zero divisor, so $abx = 0$ for some $x \neq 0$, and therefore either $a$ is a zero divisor annihilated by $bx \neq 0$ or $b$ is a zero divisor annihilated by $x$. In either case, the obvious thing to consider would be $aR+I$ or $bR+I$, but there's no obvious reason why either of these ideals should consist of zero divisors.  According to rschwieb, the answer is yes for reduced Noetherian rings.