I am trying to figure out the best title for a section of a technical document. I wish to mean tables with 12 columns. Should I write: 12-Column Grid or 12-Columns Grid ? Which is correct?
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?
Solve $\sin(z) = z$ in complex numbers Show that $\sin(z) = z$ has infinitely many solutions in complex numbers. Little Picard theorem should help, but using big Picard theorem is undesirable. Thanks a lot! I would like to know the best way to write test methods for controller extension classes that are used by Visualforce.  What is the minimum set code that I need to cover the basic operation of the controller without taking into account any custom functionality that I might add.  Here is my example class with the test method ready to be completed.  public with sharing class myExtension{      private final Account acct;      public myExtension(ApexPages.StandardController stdController){         this.acct = (Account)stdController.getRecord();     }      public static testMethod void testmyExtension(){      } } Sometimes I found that writing just "J. Doe" may lead, in a justified line with few words, to a big space between the initial and the surname. Using J.\,Doe looks ok but I would like to know if there is a general rule for spacing between initials and surnames. Using a transistor as a switch I am attaching a motor to an ATmega32 chip. I want to build an H-bridge but before doing that, I want to control the motor using the chip. I bought few NPN Transistors from RadioShack. I have 5V running though the breadboard. I attached Vcc to the Collector, Base to the Pin, and Emitter to the motor. And other wire of motor is hooked up to Ground. Unfortunately, the power coming out of the transistor is too low, the motor is barely moving (As opposed to when I hook up my motor straight to 5V). What is wrong? Do I need a different transistor or is my setup wrong? If it is the transistor, can you please recommend the right one, I am not too good at those. How to write startup script for Systemd? I have 2 graphics cards on my laptop. One is IGP and another discrete.   I've written a shell script to to turn off the discrete graphics card.  How can I convert it to systemd script to run it at start-up? cardinality of the Borel $\sigma$-algebra of a second countable space Second countability by itself doesn't restrict the cardinality of a topological space, since every set with the trivial topology is a second countable space, but it seems natural to ask whether second countability restricts the cardinality of the Borel $\sigma$-algebra of the space.  Can the cardinality of the Borel $\sigma$-algebra of a second countable space be arbitrarily big? If this is the case is there a simple construction for a second countable space with an arbitrarily big Borel $\sigma$-algebra? Why are Grothendieck's and Hartshorne's definitions of quasi-coherence equivalent? Hartshorne's Algebraic Geometry defines an $\mathcal O_X$-module $\mathscr F$ to be quasi-coherent if there is an open affine cover $(U_i=\operatorname{Spec} A_i)_{i\in\mathcal I}$ of $X$ such that each $\mathscr F\rvert_{U_i}$ is isomorphic to the sheaf of modules $\widetilde{M_i}$ associated to some $A_i$-module $M_i$. It is later proved that this is equivalent to the statement that for any open affine $U=\operatorname{Spec} A\subseteq X$, there exists an $A$-module $M$ with $\mathscr F\rvert_U\cong\widetilde M$.  On the other hand, Grothendieck's Éléments de Géométrie Algébrique defines $\mathscr F$ to be quasi-coherent if every point $x\in X$ admits an open neighborhood $U\subseteq X$ such that $\mathscr F\rvert_U$ is isomorphic to the cokernel of a morphism of free $\mathcal O_U$-modules.  Now I wonder how those definitions are equal. I tried proving it by myself, but both directions seem quite nontrivial to me... Any hints? Why are we using a biased and misleading standard deviation formula for $\sigma$ of a normal distribution? What the pointer size in 64 bits computer in C++? Is it possible to apply modifiers to multiple objects at once? You can copy modifiers from one object to other objects with CtrlL> Make links > Modifiers, or the Copy attributes addon.  However, is it possible to apply modifiers to many objects at once? When I take the pth root, the bottom of the p comes very close to the \sqrt symbol.  In the New Century Schoolbook font (which I am using), the p actually intersects the \sqrt symbol, which looks even more terrible.  Is there a clean way to fix this?  \documentclass{article} \usepackage{fouriernc} % use the New Century Schoolbook font \begin{document} When I write $a^{(1/p)}$ as $\sqrt[p]{a}$, the bottom of the $p$ touches the top of the root symbol, which looks ugly. Even when the root symbol is larger, such as with $\sqrt[p]{\frac{1}{n}}$, it still looks bad (since the tail of the p almost hits that line). \end{document} use contradiction to prove that the square root of $p$ is irrational I would like to find the correlation between a continuous (dependent variable) and a categorical (nominal: gender, independent variable) variable. Continuous data is not normally distributed. Before, I had computed it using the Spearman's $\rho$. However, I have been told that it is not right.  While searching on the internet, I found that the boxplot can provide an idea about how much they are associated; however, I was looking for a quantified value such as Pearson's product moment coefficient or Spearman's $\rho$. Can you please help me on how to do this? Or, inform on which method would be appropriate?  Would Point Biserial Coefficient be the right option? For about 2 days I'm trying to configure Ubuntu so that I can send an email, but when I type in:  echo "Subject: sendmail test" | sendmail -v my@email.com   I'll get:  postdrop: warning: unable to look up public/pickup: No such file or directory Mail Delivery Status Report will be mailed to &lt;andreigbr&gt;.   Even if I change my@email with mine, the inbox will be empty. Can somebody help me, please? I watched about 5 tutorial about how to send an email fom the command line but still doesn't work. Local html file AJAX Call and jQuery Woes In the Stack Overflow podcast #36 (), this opinion was expressed: Once you understand how easy it is to set up a state machine, you’ll never try to use a regular expression inappropriately ever again. I've done a bunch of searching.  I've found some academic papers and other complicated examples, but I'd like to find a simple example that would help me understand this process.  I use a lot of regular expressions, and I'd like to make sure I never use one &quot;inappropriately&quot; ever again. I am designing a hobby project that will run on a battery so I am trying to reduce power consumption. The SoC I am using has the ability to go into a deep sleep mode where it consumes very little power, and then can be interrupted by a falling edge on a pin.  I want to generate a high-low-high pulse when the state of a reed switch changes in either direction. I have been reading a lot and it seems that a one-shot monostable circuit can be built from a couple of transistors and an RC circuit to generate an output pulse when an edge is detected. I understand how that works. However, I can't figure out the best way to also pulse on the opposite edge.  One thought, was to just combine two one-shots that are configured to trigger on rising and falling edges respectively, and use the transistors at the output to pull the common pull-up low (kind of like a NOR gate).  Since I am a hobbyist, I am mostly using passive components. What is the "real" way of doing this in a low-power configuration using logic devices?  Edit: I figure I should mention something about the timing requirements. Normally the reed switch will be either open or closed for many seconds at minimum before changing states.  Edit: Here is an example of the circuit I was using for detecting only the falling edge:    And here is the ~20ms pulse it was producing for ~200ms button press simulating the reed switch (Channel 1 is Vbe for T1 and Channel 2 is measured at RST): Find the population size that maximizes the probability that two random samples of size $20$ will have exactly $2$ members in common Ten fish are caught in a lake, marked, and then returned to the lake. Two days later 20 fish are again caught, 2 of which have been marked.  (a) Find the probability of 2 of the 20 fish being marked if the lake has $k$ fish (assuming the fish are caught at random). (b) What value of k maximizes the probability?    Progress:     I got the (a), $$\binom{10}{2}\binom{k-10}{18}\bigg/ \binom{k}{20}$$ But how do I get part (b) where $k$ is to maximize the probability? How can I safely prototype a mains power circuit? I'm building a device that has a mains power component, which I need to prototype before building to make sure everything works. I know I need to make sure wiring—and insulation!—is thick enough to cope with the higher current, and that I need to make sure there's enough space between exposed components so there's no arcing. I'll also be stashing whatever I prototype in a nice, insulated box before applying power, and I'll be include a small-value (250mA) fuse just in case. I'll be considering everything double-insulated rather than earthed (plastic enclosure, and there's nothing I'll need to touch while it's operating).  My plan is to solder (through-hole) components directly to each other, then insulate everything (with heatshrink tubing or electrical tape) as much as possible, then use screw-type cable joiners to connect the power cable.  I know I can't use traditional prototyping tools such as breadboards and stripboard for mains voltages, but what can I use to make prototyping at mains voltages easier?  (To be clear, 'mains voltage' in my area means 230V AC +10/-6% at 50Hz; GPOs are generally rated at 10A.) Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function