"Leader's Workshop" or "Leaders' Workshop"? Leader's Workshop  or  Leaders' Workshop  Which is correct? A team of twenty leaders will be attending the workshop.
User’s Guide vs Users’ Guide I’ve been looking over what has been posted regarding the use of ’s.   I used to be a Technical Writer (years ago). The title of one of our training documents was Users’ Guide.  Once, a coworker said every time he saw that title he expected users to start coming.    Never made any sense to me, but I have to admit that the majority of us didn't understand the use of s’ in place of ’s.  What is the difference?
In what order should Tolkien's writings on Middle-earth be read? File: \boot\bcd Status: 0xc00000e9 windows 8 I am having this error for quite a few months. When the problem occurs, my windows freezes at boot time with black screen. After few reset attempts, sometimes it shows the entitled error. after few attempts, the windows starts but everything become slow. If I again restart, it works well. But at a point windows never starts and I have to reinstall it. I have already reinstalled 3/4 times in the past 2/3 months. I checked for any HDD errors with EASEUS Partition Master Home Edition with surface test, chkdisk at boot time etc for my C drive and F drive but no errors found. Startup repair also fails to solve the problem.  Windows Boot Manager   Windows failed to start. A recent hardware or software change might be the cause.   To fix the problem:  &nbsp;&nbsp;1. Insert your Windows installation disc and restart your computer.  &nbsp;&nbsp;2. Choose your language settings, and then click "Next." &nbsp;&nbsp;3. Click "repair your computer."  If you don't have this disc, contact your system administrator or computer manufacturer for assistance.   &nbsp;&nbsp;File: \Boot\BCD  &nbsp;&nbsp;Status: 0xc00000e9  My laptop's info:  Model: Dell n5110 corei5 OS: Windows 8 PRO 64 bit Anti-virus:  Avast Home edition  Additional info: My Program Files, Program Files (x86) and Users directory is located at F drive and linked with symbolic link (mklink). Also I have changed these basic locations from regedit. I also use eclipse android intelx86 emulator, virtualbox and Genymotion emulator most often. Let $R$ be a commutative ring with identity. Assume that $R$ has exactly three distinct ideals: $\{0\},I, R.$      1) Show that if $a \in R-I$, then $a$ is a unit in $R$.      2) Let $a,b\ne0$ in $I$. Show that $ab=0$. Exclude menu item from the collapse of bootstrap 3 navbar Connect to SQL Server with Windows Authentication in a different domain What is the best way to dump entire objects to a log in C#? I want to prove the definition of limit of a function from definition (1) to definition (2).  (1) A function $f(x)$ is said to have $f^*$ as the limit as $x$ tends $x^*$ iff for every sequence $\{x_n\}$ with limit $x^*$, the limit $\lim_{n-&gt;\infty}f(x_n)$ exists and is equal to $f^*$. (2) A function is said to have $f^*$ as the limit as $x$ tends $x^*$ iff for each $\epsilon &gt;0$, there exists a $\delta_\epsilon &gt;0$ such that $|f(x)-f^*| &lt;\epsilon \forall |x^*-x|&lt;\delta_\epsilon$  I am confused about the steps. Suppose I am given an $\epsilon &gt; 0$ and $x^*$. I have to show the existence of a $\delta_\epsilon &gt;0$ such that for all $x$ satisfying  $|x^*-x|&lt;\delta_\epsilon$ I can construct a sequence $x_n$ which converges to $x^*$ and $x_n=x$ for some $n \geq N$. Then I can say that $f(x_n)-f^*=f(x)-f^* &lt; \epsilon$. I think I may have made some mistake about the steps.  My main confusion is the following : Suppose I choose $\delta_\epsilon=\epsilon$. Then if a sequence $\{x_n\}$ converges to $x^*$ then $x_n=x$ (where $x$ satisfies $|x-x^*|&lt;\epsilon$) for some $n\geq N$. Now, $\lim_{n-&gt;\infty}f(x_n)$ exists implies there exists an $M$ such that $\forall n\geq M$, $f(x_n)-f^* &lt;\epsilon$. Now, I need to show that for some $n \geq M$ indeed $x_n=x$. How do I show that ? This $M$ and $N$ are different. Headless linux install How do you do a headless install of linux? No monitor, no keyboard. Machine has floopy and cd drive. Can I configure a live cd to run sshd with a preconfigured pass or something similar and manage it via that?  Instructions for fedora 8+ would be ace but anything else also welcomed.  Cheers. Let us have a surface $X$ which satisfy the equation $\omega^3 = (z-1)(z-2)\cdots (z-8)$ It's stated in the task, that there are exactly $g$ holomorphic differentials on the curve. How can I find $g$ and construct them?  The genus $g$ of this curve (I tried to use Riemann-Hurwitz theorem)  The map $X(z,\omega) \to CP^1(\omega)$ has degree 3. and we have 8 branching points with 3 leaves at each. $\chi(Domain) = DegreeOfMap * \chi(Target) - \sum_{points}(BranchingLeaves - 1) $ $2-2g = 3 *(2) - 8*2 $ hence $ 12 = 2g$ and  $g = 6$ I know that in case of hyperelliptic curve there are $g-2$ (If i'm not mistaken) holomorphic differetials and they are of sort $\frac{z^k dz}{w}$. But the form of the curve equation was heavily used. UPD: there is a branhing point at $\infty$, because if we go around all points $ 1 \cdots 8$ at once, we get that cubic root of the right side is multiplied by $\exp{4 \pi /3}$ $2-2g = 3 *(2) - 9*2 $ hence $ 12 = 2g$ and  $g = 7$ (I hope that this is true) Why Do Glueballs Have Mass, When Individual Gluons Are Massless? Floor function properties: $[2x] = [x] + [ x + \frac12 ]$ and $[nx] = \sum_{k = 0}^{n - 1} [ x + \frac{k}{n} ] $ I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:     DEFINITION Given  $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x &lt; z+1$$ and we denote it by $[x]$.   Now he asks to prove some basic things about it, such as: if $n\in \Bbb Z$, then $[x+n]=[x]+n$  So I proved it like this: Let $z=[x+n]$ and $z'=[x]$. Then we have that  $$z\leq x+n&lt;z+1$$  $$z'\leq x&lt;z'+1$$  Then $$z'+n\leq x+n&lt;z'+n+1$$  But since $z'$ is an integer, so is $z'+n$. Since $z$ is unique, it must be that $z'+n=z$.  However, this doesn't seem to get me anywhere to prove that  $$\left[ {2x} \right] = \left[ x \right] + \left[ {x + \frac{1}{2}} \right]$$  in and in general that   $$\left[ {nx} \right] = \sum\limits_{k = 0}^{n - 1} {\left[ {x + \frac{k}{n}} \right]} $$  Obviously one could do an informal proof thinking about "the carries", but that's not the idea, let alone how tedious it would be. Maybe there is some easier or clearer characterization of $[x]$ in terms of $x$ to work this out.  Another property is $$[-x]=\begin{cases}-[x]\text{ ; if }x\in \Bbb Z \cr-[x]-1 \text{ ; otherwise}\end{cases}$$  I argue: if $x\in\Bbb Z$, it is clear $[x]=x$. Then $-[x]=-x$, and $-[x]\in \Bbb Z$ so $[-[x]]=-[x]=[-x]$. For the other, I guess one could say:  $$n \leqslant x &lt; n + 1 \Rightarrow  - n - 1 &lt; x \leqslant -n$$  and since $x$ is not an integer, this should be the same as $$ - n - 1 \leqslant -x &lt; -n$$  $$ - n - 1 \leqslant -x &lt; (-n-1)+1$$  So $[-x]=-[x]-1$ The origin of the value of speed of light in vacuum I posted what I think is a clever answer to:    But then I noticed the question was posted a while ago on Mar 2.  I wanted to see what the chance of the user logging back in and seeing my answer, but the  says the user was seen on Feb 19.  I thought this was supposed to be the date they last logged in? How could they have asked a question on Mar 2 if they haven't logged in since February?  Thanks I am trying to define the probability distribution of $Z$ such as $Z = X_1\cdot X_2$ where $X_1$ and $X_2$ are two independent and identically exponentially distributed variables.  $$P(X_1=x) = \lambda e^{-\lambda x}$$ $$P(X_2=x) = \lambda e^{-\lambda x}$$  I tried something like that….  $$P(Z=x) = P(X_1 = a) \cdot P(X_2 = x/a)$$  Is this first equation correct?  $$P(Z=x) = \lambda e^{-\lambda a} \cdot \lambda e^{-\lambda x/a} = \lambda^2 e^{-\lambda (a+x/a)}$$  I doesn't feel like I found the solution as the dummy variable $a$ remains in the final result. Can you please help me solving this problem? This community wiki explains the current restrictions by the United States of America on the entry of individuals who may pose a risk of transmission of the SARS-Cov2, the virus which causes the Covid-19 disease. A friend of mine has some very old photos where the deterioration process seems to have increased in the past few years, and the owner is worried about the preservation of these old memories.  They have been stored inside an album within plastic sleeves. The owner is not aware of the display method prior to that.   I have been informed by the owner that since being in their possession, they seem to be looking worse than when first viewed and seem to be falling apart.   He is looking for the best way to preserve these photos inside an album, but not really sure what would be the best method. I have suggested for the albums as a whole to be stored in vacuum sealed bags for now, but I'm not even sure if that is good!  I am looking for any suggestions or personal experiences from anyone who has had to deal with such photos and what they did to preserve their memories. Static posts page with home.php Proof that a Combination is an integer From its definition a combination $\binom{n}{k}$, is the number of distinct subsets of size $k$ from a set of $n$ elements.  This is clearly an integer, however I was curious as to why the expression  $$\frac{n!}{k!(n-k)!}$$ always evaluates to an integer.  So far I figured:  $n!$, is clearly divisible by $k!$, and $(n-k)!$, individually, but I could not seem to make the jump to proof that that $n!$ is divisible by their product. When is the product of $n$ subgroups a subgroup? How did Luke know where to land on Dagobah?