Proving that for all $n∈\mathbb N$, if $n^2=4q+r$ for $q,r ∈ \mathbb Z$ with $0≤r<4$, then $r = 0$ or $r = 1$.
The square of an integer is congruent to 0 or 1 mod 4
How do I model the fighter's Great Weapon Fighting fighting style in Anydice? I was trying to create an AnyDice function to model the Great Weapon Fighting fighting style (which lets you reroll 1s and 2s), but I couldn't get it to work on any arbitrary dice.  I've found this one:  function: reroll R:n under N:n {    if R &lt; N { result: d12 } else {result: R} } output [reroll 1d12 under 3] named "greataxe weapon fighting"   And it works fine. But I don't know how to make the function generic so i don't need to change the d12 every time i want a different dice to reroll.  I've tried  function: reroll R:n under N:n {    if R &lt; N { result: d{1..R} } else {result: R} } output [reroll 1d12 under 3] named "greataxe weapon fighting"   but it is not giving the right probabilities. Maybe if I could fetch the die size inside the function... Microscopic interpretation of pressure in liquids Quote environment with reference at the end right When will my device get the Android 4.2 update (Jelly Bean)? What point (pt) font size are \Large etc.? In the preamble I have:  \documentclass[a4paper,11pt]{article} \usepackage{fontspec} \setmainfont{Arial}   What can be inferred about the real font pt size for the following?  \tiny, \scriptsize, \footnotesize, \small, \normalsize, \large, \Large, \LARGE, \huge, \Huge I would like to draw an arrow like this using TikZ:    This is taken from . I didn't succeed in reproducing the arrow; is somewhat beyond my (extremely limited) TikZ abilities. I don't need all the workflow stuff. I simply want to draw such an arrow at a certain place in my text, e.g. -->  \documentclass{article} \usepackage{&lt;???&gt;} \usepackage{tikz} \usetikzlibrary{&lt;???&gt;} \begin{document} Donec vehicula augue eu neque. Pellentesque habitant morbi \tikz &lt;???code for the arrow???&gt; ; tristique senectus et netus \end{document}   PS: Some explanations in your solution would be much appreciated, otherwise it will be difficult to customize it. Using a naked black hole as a mirror? Tricks and tips for finding information in man pages Does anyone have any tricks and tips for finding information in man pages? highest power of prime $p$ dividing $\binom{m+n}{n}$ How to imitate commercial fried chicken? I'm trying to develop my own "secret" fried chicken recipe, something that is similar to Popeye's or Church's. I'm a LONG ways from getting there. Here's what I did in my last experiment:   Chicken was marinated in buttermilk with Cajun spices and Tabasco sauce for 24 hours I used a deep fryer and canola oil. 175C. Prior to "dressing" the chicken, I let the excess buttermilk brine drip off the chicken. I then seasoned it with Cajun seasoning I made a batter using an egg, self rising flour, milk, baking powder I also tried using just plain flour for breading   Results:   Using the batter, the chicken came out looking burned and too smooth (didn't have that crunchy look). It burned too much that I couldn't finish cooking the chicken. Using the flour, the chicken came out with a thin crust, somewhat crispy, but also too burnt. Dipping the chicken in the batter AND then flour gave a bulky &amp; crunchy look. However, again, the result looked too burnt.   Modifications:  I then changed the the breading. I used bread flour and corn starch. I dipped the chicken in the batter again, then used the new breading. The bread flour was a bit lighter. And the corn starch was even more lighter. But they were still burn looking (just lighter shades). I also dropped the temperature to 160C.  Results:  Burnt chicken whose crust wasn't really sticking to the skin/meat.  Questions:  Any suggestions on what I can do to improve? I'm guessing something in the batter is causing the burnt coat. What flour should I use? What can I do to make the coat stick to the skin/meat? Any other suggestions on what to do? Understanding complete separation for logistic regression Why does logistic regression not converge for a linearly separable data set?   For linear separable data sets the model parameters go to infinity when mimizing the error function (according to Bishop2006, Pattern recognition and machine learning, section 4.3.2). I don't understand why. It must be connected to setting the score function, i.e. the derivative of the error function, to zero.  The error function is given by $E(\mathbf{w})$ as shown in the figure below. Where $\mathbf{t}$ is the target variable vector of the training set and $\mathbf{w}$ the parameter / coefficient vector. The posterior probabiltiy for class 1 is given by  $$y(n) = p(C_1|\mathbf{x}_n) = \sigma(\mathbf{w}'\mathbf{x}_n).$$ Where sigma is the logistic sigmoid function and $\mathbf{x}_n$ is the feature vector for instance $n$ including the predictors.     My second question to logistic regression is if the to posterior probabilities $p(C_1|\mathbf{x}_n)$ and $p(C_2|\mathbf{x}_n)$ sum to one? How to make equally sized windows in corners of the screen? I was trying to solve internet problems, and seem to have made things worse: Now when I start the computer it gets stuck on "Booting system without full network configuration".   Does anyone know how to make it unstuck?   Alternatively: It got messed up when I changed the contents of /etc/network/interfaces, so presumably would work again if I changed it back. How can I edit this file during boot?   I'm using Ubuntu 12.10. It is quite easy to calculate the total number of functions from a set $X$ with $m$ elements to a set $Y$ with $n$ elements ($n^{m}$), and also the total number of injective functions ($n^{\underline{m}}$, denoting the falling factorial). But I am thinking about how to calculate the total number of surjective functions $f\colon X \twoheadrightarrow Y $.  The way I thought of doing this is as follows: firstly, since all $n$ elements of the codomain $Y$ need to be mapped to, you choose any $n$ elements from the $m$ elements of the set $X$ to be mapped one-to-one with the $n$ elements of $Y$. This results in $n!$ possible pairings. But the number of ways of choosing $n$ elements from $m$ elements is $\frac{m!}{(m-n)!\,n!}$, so the total number of ways of matching $n$ elements in $X$ to be one-to-one with the $n$ elements of $Y$ is $\frac{m!}{(m-n)!\,n!} \times n! = \frac{m!}{(m-n)!}$.  Now we have 'covered' the codomain $Y$ with $n$ elements from $X$, the remaining unpaired $m-n$ elements from $X$ can be mapped to any of the elements of $Y$, so there are $n^{m-n}$ ways of doing this. Therefore I think that the total number of surjective functions should be $\frac{m!}{(m-n)!} \, n^{m-n}$.  Is this anything like correct or have I made a major mistake here? Difference between Analytic and Holomorphic function A function $f : \mathbb{C} \rightarrow \mathbb{C}$ is said to be holomorphic in an open set $A \subset \mathbb{C}$ if it is differentiable at each point of the set $A$.  The function $f : \mathbb{C} \rightarrow \mathbb{C}$ is said to be analytic if it has power series representation.  We can prove that the two concepts are same for a single variable complex functions. So why these two different terms? Is there any difference between these two concepts in general, please give example.  Thank you for your help. For a CDF that is not strictly increasing, i.e. its inverse is not defined, define the quantile function   $$F^{-1} (u) =\inf \{x: F(x) \geq u \},\quad 0&lt;u&lt;1. $$  Where U has a uniform $(0,1)$ distribution. Prove that the random variable $F^{-1} (u)$ has cdf $F(x)$.  In case of a strictly increasing CDF the proof is quite easy because the inverse is defined. Define $X=F^{-1} (u)$  $$ P\left[X&lt;x \right]=  P \left [F^{-1} (U) \leq x \right]= P \left[U \leq F(x) \right] =F(x)  $$  But how do I accomondate the nondecreasing CDF whose inverse is given by the quantile function? I am a begginer so any help is welcome. Thank you. A local IP address (RFC 1918) on my network that is not … local I've noticed an IP address today on my network (192.168.15.1) which when visited actually goes to a page that has a router webconfig page (Not my router)  The puzzling thing is that i tried traceroute 192.168.15.1 which said that the distance between me and that host is 9 hops, whereas the distance between me and any other computer on my home network is 2 hops away (My computer --> Router --> Host)   Can someone please explain what's going on here ?  tracert output:  Tracing route to 192.168.15.1 over a maximum of 30 hops    1    &lt;1 ms    &lt;1 ms    &lt;1 ms  192.168.1.1   2    47 ms    51 ms    49 ms  [163.121.171.123]   3     *        *        *     Request timed out.   4   128 ms     *        *     163.121.211.134.dnet.net [163.121.211.134 ]   5     *      134 ms   126 ms  10.37.242.18   6    94 ms    90 ms    95 ms  10.35.2.157   7   178 ms   254 ms   206 ms  10.32.8.107   8    85 ms    57 ms   318 ms  10.35.0.186   9   215 ms   241 ms   230 ms  192.168.15.1  Trace complete. Query to find GUID's "not in" null containing subquery returns no results How can I calculate the age of a person in year, month, days? When using the Profile Tool plugin I get a graph that looks like the 2nd derivative of what I want. That is, it looks like the rate of change of the slope of the elevation profile. I just want the elevation profile! The data is present for good 1' contour lines. Why is the Profile Tool interpreting the data in such a bizarre way? What am I missing? And what are the Min and Max units?