Function pow() does not work correctly when using big numbers?
Is floating point math broken?
Disable COPY drag and drop in Windows Explorer in Windows 7 Chrome's default search engine is .  How we can force chrome to use  without https?  I need a solution that allows proxies to rewrite my URLs, but https currently makes this impossible... This question is a possibly harder version of: .  Question.  Let $f,g :\mathbb R\to\mathbb R$, such that \begin{align} f(x-y)=f(x)\, g(y)-f(y)\, g(x), \tag{1}\\ g(x-y)=g(x)\, g(y)+f(x)\, f(y), \tag{2} \end{align} for all $x,y \in \mathbb{R} $. If $g$ is continuous at $x=0$ and not identically zero, then there exists an $\alpha\in\mathbb R$, such  $$ f(x)=\sin \alpha x\quad\text{and}\quad g(x)=\cos \alpha x. $$ Is there a pair of discontinuous $f$ and $g$ satisfying $(1)$ and $(2)$?  Update. If $\ell :\mathbb R\to\mathbb R$ is a linear functional over $\mathbb Q$ (i.e., $\ell(qx+ry)=q\ell(x)+r\ell(y)$, for all $x,y\in\mathbb R$ and $q,r\in\mathbb Q$), then $$ \sin\big(\ell(x)\big), \quad \cos\big(\ell(x)\big), $$ satisfy $(1)$ and $(2)$. Discontinuous such functionals do exist, and they are obtainable using Zorn's Lemma (equivalently the Axiom of Choice.) This takes care of the second question. S tabular with alignment I'd like to create a picture like the one with TikZ:    I've already created the electrical part using circuitikz. I'd like to add the part on the right (the motor's shaft with arrows), but don't know how to start, actually. Is there a way to draw 3D shapes, maybe? Or should I draw circles and rotate them?  \documentclass{article} \usepackage{tikz} \usetikzlibrary{intersections} \usepackage{circuitikz} \usepackage{siunitx}  \begin{document}  \begin{figure} \centering \begin{circuitikz} \draw (0,0) to [V, v=$v(t)$, *-*] (0,2)      to [R, l=$R_d$] (2,2)     to [L, l=$L_d$] (4,2)     to [V, v&lt;=$e(t)$] (4,0) to [generic] (4,2) -- (4,0) -- (0,0); \end{circuitikz} \end{figure}  \end{document}   What I have now:   What I'm still missing: Aligning multiple points to line in QGIS? I would like to align multiple points to a line, or lines within a layer, using a specified tolerance or buffer around the line objects. Please refer to the sample sketch attached.  For the sake of this example, the points closest to the line in the BEFORE picture are within 5 map units of the line, while the outermost points are over 10 map units away. I would like to snap the closest points onto the nearest line, using a tolerance of 5 map units to achieve the result in the AFTER picture. How to add a gif file to my LaTeX file? Related:          I have an image file -- a gif file, which is the graph of a function. How do I add it to a LaTeX file? Will it come out OK in the subsequent pdf file? I am studying the function and characteristic of an op-amp to build a line-following robot with the schematics below. However, in many tutorials, the way to differentiate an inverting and non-inverting op-amp is by looking at the connection to ground via the inverting or non-inverting terminal.  However, in the schematics, the op-amp is considered to have connections to ground, thus in this case, how do I differentiate it then?  I may have understood the whole concept wrongly. Pls help!    EDIT: Please ignore the switch as it may have been drawn wrongly by the supplier. Let $S$ be the collection of all groups. Deﬁne a relation on $S$ by $G \sim H$ iﬀ $G ≈ H$. Prove that this is an equivalence relation. So $S$ is partitioned into isomorphism classes.   Proof: Let $S$ be a relation on S defined by $G \sim H$ iﬀ $G ≈ H$. Show reflexive, symmetric, transitive. It is obvious that $G≈G$ so $G\sim G$.  Assume $G\sim H$, thus $G≈H$, which implies that $H≈G$, so $H\sim G$, thus symmetric.  Assume $G\sim H$ and $H\sim K$, for $K$ a group in $S$. Thus $G≈H$ and $H≈K$. (I have already shown that if $G≈H$ and $H≈K$, then $G≈K$). From earlier this implies that $G≈K$, so $\sim$ is an equivalence relation.  Thus $S$ is partitioned into isomorphism classes. QED. Empty intersection and empty union Blank question bodies in Android app Version 1.0.58  When viewing questions that contain multiple code blocks, the question is blank for a few screens. Answers and comments are OK.   I tried clearing application data, and rebooting they phone, no change. Mobile sites work perfectly.   Example:      Also, now that I posted the pictures and links, this question also went blank How are the file metadata stored in Windows? (I'm using Windows XP but I guess it's similar in all the recent versions of Windows.)  When you create for example a new empty text document, you'll find in its properties that it has size of 0 bytes. Zero bytes means no information. No data. But still, the file has some name, it can still carry the dates of last access, modification and creation. It carries the information whether it's hidden file or not, whether it's read-only or not...  So where are all the metadata stored? I have a daytime raster of my study area. I have 20+ counters in the study area that count "stuff" with a timestamp. Using  I determined the type of ambient light for each collected data point.   What I'd like to do:   Add a grey/semi transparent overlay to reflect the ambient light (Dusk, Sunrise, day, sunset, and dark). The end goal: produce a time-based visualization of the data collection that highlights the time of the day. Or be able to produce a snapshot at a certain time with the proper shaded overlay. Think of this as this the image: Windows Unable to Delete ._. File Any way to tell apart a CPU defect from a mainboard defect? I have a fairly modern desktop computer, an AMD Athlon II X2 based silent PC, that does not work.  When turned on, it will start physically (all fans are rotating, disks start up...) but not give a signal on any of the graphics ports (DVI, VGA, and HDMI, I tried all three). Also, the reset button does not seem to have any effect.  I have stripped the mainboard bare of all SATA connections, extension cards, and the one 2GB RAM chip to eliminate them as the problem source, but to no avail, so I think it's fair to assume it's either the mainboard or the processor that are at fault.  However, I have neither a replacement mainboard, nor a replacement processor handy to identify which one is broken. The cause of the defect is unknown, so for all I know, it could be both. Therefore, I'm reluctant to buy replacement hardware blindly before knowing more.   Is there any way to further diagnose (or at least get some indication) which component is broken without buying replacement hardware? IE must close for event with jQuery to work How do you file a bug report for Windows? I have come across an issue that I feel is a  in Windows 7. What is the official process for submitting an issue to Microsoft from a consumers perspective as an end user of Windows? Choice function for collection of arbitrary finite sets. AC required? Subgroups of $U(n)$ isomorphic to a direct sum of cyclic groups Let $U(n)$ to be the set of all positive integers less then $n$ and relatively prime to $n$. Then $U(n)$ is a group under multiplication modulo $n$.  A. Find integer $n$ such that $U(n)$ contains subgroup isomorphic to $Z_5\oplus Z_5$.  B. Show that there is an $U(n)$ containing a subgroup isomorphic to $Z_3 \oplus  Z_3$, but not isomorphic to $Z_4 \oplus  Z_4$. Almost Everywhere Convergence versus Convergence in Measure