Given that a generic $n$-gon ($n\geq3$) is inscribed in a circle in $\mathbb{R}^2$, we know the lengths of all its edges $a_1,a_2,\cdots,a_n$. Are there formulae to calculate its area (under Euclidean metric)?  For $n=3$, let $p=\frac{\sum_{i\in [n]} a_i}{2}$, we have the Heron's formula: $$A=\sqrt{p(p-a_1)(p-a_2)(p-a_3)}$$ which is the special case of the following $n=4$ formula ($a_4=0$): $$ A=\sqrt{(p-a_1)(p-a_2)(p-a_3)(p-a_4)}.$$ Are there any known result for $n\geq 5$?  If not, any proof that certain formulae don't exist?
Let's say we are given a set of positive reals, and we're told that these are the edges of a convex  $n$-gon, and we must compute it's area.   For $n = 3$ there is the famous Heron's formula:  $$A = {1\over 4}\sqrt{4a^2b^2 - (a^2 + b^2 - c^2)^2}$$  For $n = 4$ there is Brahmagupta's formula:  $$A = {1\over 4}\sqrt{(a^2 + b^2 + c^2 + d^2)^2 + 8abcd - 2(a^4 + b^4 + c^4 + d^4)}$$  Is there a generalization for arbitrary $n$? Does it even exist?
Refresh Icon Cache Without Rebooting Normally, to refresh the icon cache in Windows, we have to reboot.  Is there a way to refresh the icon cache in Windows 7/8 without rebooting? Calculate expected values using logistic regression coefficients The outcome of the logistic model is binary (0;1) - intervention that occurs in hospital (3838 interventions in 51096 cases). I am modeling the effect of age groups (60 to 74; 75 to 89 and 90+) and sex (female) on the outcome. My reference age group is male aged 18-59.  My logistic regression coefficients look like this:  intercept  -4.313 female      0.5117 age 60-74   0.4081 age 75-89   1.1761 age 90+     1.5612.   Corresponding odds ratios:  female      1.668 years60_74  1.504 years75_90  3.242 years90     4.764   Now I am trying to obtain a probability of an outcome = 1 for each case. What is the right way to do this? Box Cox Transforms for regression I'm trying to fit a linear model on some data with just one predictor (say (x,y)). The data is such that for small values of x, the y values give a tight fit to a straight line, however as x values increase, the y values become more volatile. Here is an example of such data (R code)  y = c(3.2,3.4,3.5,3.8,4.2,5.5,4.5,6.8,7.4,5.9) x = seq(1,10,1)   I'm curious to know if there exists any power transform (Box cox perhaps?) that lets me get a better fit for the data than simply doing a linear fit as shown below.  fit = lm(y ~ x) Refactoring RadioGroup setter code? Expanding a parent  to the height of its children I have a page structure similar to this:  &lt;body&gt;   &lt;div id="parent"&gt;     &lt;div id="childRightCol"&gt;       /*Content*/     &lt;/div&gt;     &lt;div id="childLeftCol"&gt;       /*Content*/     &lt;/div&gt;   &lt;/div&gt; &lt;/body&gt;   I would like for the parent div to expand in height when the inner div's height increases.  Edit: One problem is that if the width of the child content expands past the width of the browser window, my current CSS puts a horizontal scrollbar on the parent div.  I would like the scrollbar to be at the page level. Currently my parent div is set to overflow: auto;  Can you please help me with the CSS for this? Invariant Subspace of Two Linear Involutions I'd love some help with this practice qualifier problem:  If $A$ and $B$ are two linear operators on a finite dimensional complex vector space $V$ such that $A^2=B^2=I$ then show that $V$ has a one or two dimensional subspace invariant under $A$ and $B$.    Thanks! Given any $10$ consecutive positive  integers , does there exist one integer which is relatively prime to the product of the rest ? How do I prompt for Yes/No/Cancel input in a Linux shell script? I want to pause input in a shell script, and prompt the user for choices. The standard Yes, No, or Cancel type question. How do I accomplish this in a typical bash prompt? The title is kind of obscure. What I want to know is if this is possible:  string typeName = &lt;read type name from somwhere&gt;; Type myType = Type.GetType(typeName);  MyGenericClass&lt;myType&gt; myGenericClass = new MyGenericClass&lt;myType&gt;();   Obviously, MyGenericClass is described as:  public class MyGenericClass&lt;T&gt;   Right now, the compiler complains that 'The type or namespace 'myType' could not be found." There has got to be a way to do this. ModelBuilder not adding output feature class to Table of Contents? I'm trying to create a tool in ModelBuilder that will take a new point from a user, create a feature class for this point and display it on a map.  What I've done is:   I've created a feature class for a point-like feature in ArcCatalog. I've prepared a really simple model in ModelBuilder, that starts with a feature set variable (which takes the feature class as a schema and symbology file). The variable is marked as a parameter. Next I'm using the Copy Features tool that takes the variable as input and saves it as a feature class in my scratch workspace. The output feature class is marked as a parameter with the 'add to display' option.   After the module is run I get a shapefile in my scratch workspace. I can add it manually to TOC and it contains the selected point. However no matter what I do, I can't display it in TOC automatically after the model completes.  There is a known issue described in several places that doesn't allow to display newly created layers in TOC when model finishes with the 'add to display' option. However there's a quick workaround, which is to mark the output layer as parameter. In my case it doesn't work.  I'm using ArcGIS for Desktop 10.1 on Windows 7 x86_64. I read  and thought that would easily be solved (not that it isn't solvable without) if one could write:   @Override public String toString() {     return super.super.toString(); }   I'm not sure if it is useful in many cases, but I wonder why it isn't and if something like this exists in other languages.  What do you guys think?  EDIT: To clarify: yes I know, that's impossible in Java and I don't really miss it. This is nothing I expected to work and was surprised getting a compiler error. I just had the idea and like to discuss it. When learning a new subject, I would frequently use lecture notes found somewhere in the Internet. When writing a paper (or a master thesis, as in my case, but the rules should be similar, I believe) one should give some reference for used results which are not common knowledge, if I understand correctly. This make me wonder: what do I do if I want to reference a result I found in some notes?  The natural thing to do would be to just add these notes to bibliography. What format would be preferable for this? Note that there will generally not be much publishing information, perhaps not even a definite year and place. (A BibTeX template would be .)  Secondly, is it OK to cite such materials as a reference? General solution of expected value of E(f(X))? This is maybe a trivial question I came up while solving a few examples and understanding Markov/Chebyshev inequalities and subsequently in evaluating Chernoff bounds. Suppose $X$ is a random variable with some distribution $p$ and we need to evaluate $\mathbb{E}[f(X)]$. Here, $f(X)$ may or may not have the same distribution as $X$. In my case, if $f(X) = e^{sX}$, where $s&gt;0$. I would work out this problem by letting $Z = e^{sX}$ and   $\mathbb{E}[f(X)] = \mathbb{E}[Z] = \int_{\infty}^{\infty}z \hspace{0.2em}p(z)dz$.   Now I find the distribution of $Z$ as: $P\{ Z \leq z \} \implies P\{ e^{sX} \leq z \} \implies P\{ X \leq \frac{\log(z)}{s}\} = \int_{-\infty}^{\frac{\log(z)}{s}}x.p(x)dx $  thereby substituting this into the expectation one gets $\mathbb{E}[Z] = \int_{\infty}^{\infty}z\hspace{0.2em} \left(\int_{-\infty}^{\frac{\log(z)}{s}}x\hspace{0.2em}p(x)dx\right)^{\prime} dz$  According to the Law of Unconscious Statistician , this approximates distribution of $Z$ as $p$. I understand the backward proof through calculus, I don't follow the measure theory approach. For example I don't follow how this works if say $X \sim Gamma$. Is there a source I can read through for someone with no measure theory background? "rsync: failed to set permissions on ..." error with rsync -a or -p option How to calculate overlap between empirical probability densities? I'm looking for a method to calculate the area of overlap between two kernel density estimates in R, as a measure of similarity between two samples. To clarify, in the following example, I would need to quantify the area of the purplish overlapping region:  library(ggplot2) set.seed(1234) d &lt;- data.frame(variable=c(rep("a", 50), rep("b", 30)), value=c(rnorm(50), runif(30, 0, 3))) ggplot(d, aes(value, fill=variable)) + geom_density(alpha=.4, color=NA)     A similar question was discussed , the difference being that I need to do this for arbitrary empirical data rather than predefined normal distributions. The overlap package addresses this question, but apparently only for timestamp data, which doesn't work for me. The Bray-Curtis index (as implemented in vegan package's vegdist(method="bray") function) also seems relevant but again for somewhat different data.  I'm interested in both the theoretical approach and the R functions I might employ to implement it. My laptop was working perfectly yesterday, and now today I can't log in.  At first, I was only getting a black screen. However, after following instructions in some questions here on AskUbuntu, I first reinstalled lightdm, and then I had to change the ownership of the file ~/.Xauthority to be my user name. Now, I get the log in GUI screen.  However, once I enter my username, it flickers and then comes back to the login GUI. It does not matter if I use Gnome, Unity, or Gnome-Flashback.  I don't know why lightdm needed to be reconfigured, or how ~/.Xauthority got changed, but in any case, what is still standing in my way, preventing me from logging in?  Update: I have tried deleting the .Xauthority file, and the .profile files in my home directory. It has not changed anything.  Logging in as guest also fails to work.  The following commands did not work:  mv ~/.config ~/.config.BAK mv ~/.cache ~/.cache.BAK   Inside .xsession-errors in my home directory, it says:  Gdk-CRITICAL: gdk_x11_display_get_xdisplay: assertion 'GDK_IS_DISPLAY (display)' failed Is it OK to call a professor by his first name when he/she signs emails by only first name? If a professor in a North American country presents his- or herself by his/her first name in email messages, does this mean that students can refer to him/her by his/her first name? Or is this generally not a good idea, unless the professor has explicitly mentioned that he/she can be referred to by first name? I've noticed that most professors, who prefer to be referred to more formally, do not sign their emails with just their first names, but usually initials or first and last name. What is the probability that a man likes pink? How do I resolve this installation error? I am getting the following error while installing Drush with composer require drush/drush.     Using version ^8.1 for drush/drush   ./composer.json has been updated   Loading composer repositories with package information   Updating dependencies (including require-dev)   Your requirements could not be resolved to an installable set of packages.            Problem 1   - drush/drush 8.1.0 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.1 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.2 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.3 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.4 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.5 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.6 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.7 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.8 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.9 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - Conclusion: don't install drush/drush 8.1.12   - Conclusion: don't install drush/drush 8.1.11   - Conclusion: don't install drush/drush 8.1.10   - Conclusion: don't install drush/drush 8.x-dev   - Installation request for drupal/drupal [No version set (parsed as 1.0.0)] -> satisfiable by drupal/drupal [No version set (parsed as 1.0.0)].   - Conclusion: remove phpdocumentor/reflection-docblock 3.1.1   - Installation request for drush/drush ^8.1 -> satisfiable by drush/drush [8.1.0, 8.1.1, 8.1.10, 8.1.11, 8.1.12, 8.1.2, 8.1.3,   8.1.4, 8.1.5, 8.1.6, 8.1.7, 8.1.8, 8.1.9, 8.8.x-dev, 8.x-dev].   - Conclusion: don't install phpdocumentor/reflection-docblock 3.1.1   - drush/drush 8.8.x-dev requires phpdocumentor/reflection-docblock ^2.0 -> satisfiable by phpdocumentor/reflection-docblock [2.0.0,   2.0.0a1, 2.0.0a2, 2.0.0a3, 2.0.1, 2.0.2, 2.0.3, 2.0.4, 2.0.5].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0a1, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0a2, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0a3, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.1, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.2, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.3, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.4, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.5, 3.1.1].   - Installation request for phpdocumentor/reflection-docblock (locked at 3.1.1) -> satisfiable by phpdocumentor/reflection-docblock [3.1.1].      Installation failed, reverting ./composer.json to its original   content. An example of a sequence of Riemann integrable functions $(f_n)$ that converges pointwise to a function $f$ that is not Riemann integrable. I have been given the example $$ f_n(x)= \begin{cases} 1, &amp; \text{if}\; x\in\{(x_k)_{k=1}^{n}\}, \\ 0, &amp; \text{otherwise.} \end{cases} $$ Here the sequence $(x_k)_{k=1}^{\infty}$ is the sequence that "enumerates" elements of $\mathbb{Q}$. The sequence of functions converges pointwise to $$ f(x)= \begin{cases} 1, &amp; x\in\mathbb{Q}, \\ 0, &amp; x\notin\mathbb{Q}, \end{cases} $$ which isn't Riemann integrable. However, I am not quite convinced that $f_n$ is integrable. I would like some help in seeing how $f_n$ is Riemann integrable. Also, I would like more examples of sequences of Riemann integrable functions $f_n$  that converge pointwise to a function $f$ that is not Riemann integrable.