Random selection of polygons in ArcMap?
How to randomly subset X% of selected points?
Are quotes needed for local variable assignment? Can I safely omit quotes on the right side of a local assignment?  function foo {     local myvar=${bar}     stuff() }   I'm mainly interested in bash, but any info on corner cases in other shells are welcome. Asynchronous shell exec in PHP $n$ choose $k$ where $n$ is less than $k$ I am working on parameter estimation and one of the estimators involves a summation of $_nC_k$ ($n$ choose $k$) expressions. For some iterations, I need to compute expressions like $_0C_1$, $_0C_2$, etc.  In general how do we compute $_nC_k$ when $n$ is less than $k$? Do we still use the formula $\frac{n!}{(n-k)!k!}$ and use the gamma function to compute the negative factorial?  Thanks! How to return only the Date from a SQL Server DateTime datatype In set theory, if natural numbers are represented by nested sets that include the empty set, how are the rest of the real numbers represented as sets?    Thanks for the answers. Several answers basically said for irrational numbers that  A Dedekind cut is a pair of sets of rational numbers $\{L, R\}$. The set of real numbers is defined to be the set of all Dedekind cuts, where a Dedekind cut is a pair of sets of rational numbers $\{L, R\}$ which have no elements in common, and where all the elements of $L$ are less than any element of $R$.  Each Dedekind cut is a real number. This is where I have a problem - surely that can’t be correct. The set $L$ is a set of all rationals, and there must be a rational in the set $L$ that is greater than all other rationals in that set, even if we have no method of determining it. And similarly, there must be a rational in the set $R$ that is less than all other rationals in that set, even if we have no method of determining it. If every irrational number has a corresponding set $L$, then each irrational number has some such corresponding largest element of that set $L$, and then each irrational number has some corresponding rational number. And that would mean that the irrational numbers are countable. So, with Dedekind cuts,  the only conclusion is that there must be irrational numbers $x$ which are either greater or lesser than some irrational cut $y$ of the rationals, and between $x$ and $y$ there is no rational number.  But that is impossible, so that the Dedekind cuts cannot be the correct representation of the real numbers.  Surely the problem with Dedekind cuts is in using sets of rationals that include all rationals up to a certain rational. But there is an alternative method of representing irrationals can be defined in terms of infinite sets of rational numbers. For example, in binary notation, the non-integer part of $\pi$ is $.00100100\ 00111111\ 01101010\ 10001$. You define a set by: if the nth digit is a $1$, then  the natural number $n$ is in the set. And then we have that, for the real numbers between $0$ and $1$, that the set of real numbers is simply the set of all subsets of natural numbers. Each subset corresponds to some real number between $0$ and $1$.  And in this way, all real numbers can be considered to be some set based only on nested sets of the empty set.  But I still haven’t got a satisfactory answer for how negative numbers can be represented in terms only of sets containing the empty set. Any ideas? Why is filling a balloon from your mouth much harder initially? Why is it that when you first fill up a balloon, it's hard to get air through, but after inflating it a bit, it becomes much easier to further inflate the balloon? How do I verify that a string only contains letters, numbers, underscores and dashes? I know how to do this if I iterate through all of the characters in the string but I am looking for a more elegant method. How do we know for certain that space is expanding? How do we know for certain that space is expanding?  Let's say that in the year 1950, we observe that galaxy 1 is 5 billion light years away from us and galaxy 2 is 10 billion light years away from us, putting both galaxies at a distance of 5 billion light years from each other. Then in 2013, we observe that they are now 7 billion light years away from each other so we conclude space is expanding.  We see that galaxies are move away from each other but how does that prove that space is expanding? Could this illusion of expanding space simply be due to something larger in mass pulling the farther galaxy 2 away from galaxy 1? This larger mass could be accelerating the velocity of galaxy 2 faster then it's accelerating the velocity of galaxy 1. How to write the word 'smart phone' in a research article? Smart phone Smartphone Smart-phone   I am really looking for some help regarding hyphen (-) issue in text found in research article. Is there any standard method how to use it? What effect does the Barr body have, in relation to female Turner syndrome? Why do persons with  have  developmental abnormalities, when normal XX-females do not, even though they only have 1 active X chromosome? From what I know, one X-chromosome is inactivated into a Barr body in normal XX-females, but these obviously don't show the symptoms of Turner syndrome. What function does the inactive chromosome have? When searching on a site with  enabled (), I saw some  rendering issues: (and currently it's also in the )        Even though the LaTeX rendering is fine when opening the :        The LaTeX code is:  $f(n) = \begin{cases} 0,  &amp; \text{$ \omega&lt;0, \omega=1, \omega&gt;3 $} \\[2ex] -\omega + 1,  &amp; \text{$ 0\le\omega&lt;1 $} \\[2ex] \omega^2 - 1,  &amp; \text{$ 1&lt; \omega \le 2 $} \\[2ex] 3 \omega,  &amp; \text{$2 &lt; \omega \le 3 $} \end{cases}$ Unexpected output of xargs What Is Exponentiation? Compute $\mathbb{P}\{ W_t < 0 \, \, \text{for all} \, \, 1 < t < 2\}$ for a Brownian motion $(W_t)_{t \geq 0}$ Let $(W_t)_{t \ge 0}$ be a standard one-dimensional Brownian motion. Calculate $$\mathbb{P}\{ W_t &lt; 0 \, \, \text{for all} \, \, 1 &lt; t &lt; 2\}.$$ I can only think that this will be conditional on $W_1$. Please tell how to proceed? Wired connection not working Ubuntu 12.04 How are the measurements used by Nielsen evolving to meet new methods of content delivery and viewing? I was interested to find out that in the 1950s, Nielsen used devices in the homes of the families that were surveyed to record their viewing preferences, and only later introduced their system of using a diary to record habits:  ...in 1950...This information was collected on a device that was attached to a television that recorded what was being watched.  In 1953, the company began sending out diaries to a smaller sample of homes (“Nielsen families”) within the survey to have them record what they had watched from   So, presumably as multiple different viewers in the home were considered in the sample, the means through which the ratings were recorded changed. Given that so many people are able to record shows digitally and watch when convenient and that so many view their favorites on outlets such as iTunes, how have the Nielsen ratings systems adapted (or how does the company plan to adapt them) to capture our modern viewing preferences? While a DVR-recorded show still technically makes its original &quot;timeslot&quot; relevant to the equation, will Nielsen consider timeslot-specific data a hindrance to comparisons between those shows viewed on cable and the same show viewed on iTunes? Could a constant function be considered increasing? Given the definition of an increasing function as a function $f:E\to\mathbb{R}$, where $E$ is nonempty such that $x_1,x_2\in E$ and $x_1&lt;x_2$ imply $f(x_1)\leq f(x_2)$, then couldn't, for example, $f(x)=-1$ considered increasing? Since the inequality is not a strict inequality. I used to be able to go to Accounts and Import → Add another email address you own and add an email account that's only a forwarding address. That is, it's not a mailbox and I don't have SMTP access to it.  Gmail would send me a confirmation email to that address, I'd confirm it's mine, and done. Now it only gives me the option to add SMTP credentials.  Did they stop supporting the previous method? Is there any other way I can get that option back? Regarding the `book`, `report`, and `article` document classes: what are the main differences? Is it time to be rid of automatic down votes on certain closed questions? Right now,  carry automatic downvotes.  The  for this was largely to enforce specific quality bans in part because people were not down voting questions. Now that question down votes are free on questions and it seems to be pretty commonplace for question quality bans to trigger, even without the automatic votes from closure, and we have completely revamped the closure system. Is it time to reevaluate whether or not these automatic votes are still a good idea?  What benefits do the automatic votes provide and have they effectively run their course?