$X$, $Y$ and $Z$ are three random variables where their pairwise correlations are the same, i.e., $corr(X,Y)=corr(Y,Z)=corr(X,Z)=\rho$ what are the possible values of $\rho$?  I was asked this in an interview (data analyst).
There are three random variables, $x,y,z$. The three correlations between the three variables are the same. That is,  $$\rho=\textrm{cor}(x,y)=\textrm{cor}(x,z)=\textrm{cor}(y,z)$$  What is the tightest bound you can give for $\rho$?
Can someone provide a proof that every quadratic residue, when working in $\mathbb Z_p$, where $p$ is a prime, has exactly two roots? Indeed, there cannot be only one root as for any $a^2$, we know $a$ and $-a$ are both roots. So then we need only show that there cannot be more than two- does this follow from some argument for the number of roots a quadratic can have modulo a prime? I'm not well-acquainted with the theorems pertaining to prime moduli. How to remove/hide the recycling bin and All site content from the quick launch or remove the quick launch altogether? Ho can I remove the recycling bin and the 'All Site Content' link from the quick launch or remove the quick launch altogether How to reinstall Ubuntu keeping my data intact? Show that $\{\cos\ n:n \in \mathbb{N}\}$ is dense in $[-1,1]$ by using the fact below:      Suppose $x$ is irrational. Then there exists $p_n,q_n \in \mathbb{Z}$   such that $\bigg|x -\frac{p_n}{q_n}\bigg| &lt; \frac{1}{q_n^2}$   I have no idea on how to apply the fact above to show the set is dense in $[-1,1]$. Can anyone help me? How to reduce AWS EBS root volume size? Example to prove that $ C^1[0,1] $ is not a Banach space for the uniform norm? The space $ C^1[0,1] $- the space of all continuously differentiable functions on $ [0,1]$ is not a Banach space with respect to the sup norm,$ \|.\|_{\infty} $ since the uniform limit of a continuously differentiable function need not be differentiable.  How can I illustate this statement using a counter example? Can I use $ f_{n}=\frac1 n \sin nx $ as a counter example?  Also, is $ C^1[0,1] $ is same as the space given by $ X=\{f\in C^1[0,1]:f(0)=0 \}$. Can I use the same example to show that this is not a Banach space?  More help is appreciated! Thanks! How to change the chapter number to display as text (One, Two, ...)? In LaTeX, how do I change the chapter number to display as text? eg:  CHAPTER ONE not CHAPTER 1;  CHAPTER THIRTEEN not CHAPTER 13. Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$? Can 12 teams in 6 disciplines play 6 rounds without repetition? Making voltmeter accepting bipolar input voltage, using a microcontroller I want to build a very simple voltmeter using ATMEGA328 ADC. I can successfully measure up to 20v (using a voltage divider), but the problem is, I don't understand how to measure the voltage properly, if negative and positive terminals are switched.   Only way I could think of is adding another ADC input which is responsible for measuring the 'inverted voltage'. The join both ADC inputs, but add diodes in between, so one channel works with (+-) and the other channel works with (-+) and then 'merge' the input from both channels using code. Then I should be able to measure things like sin wave for example (where polarities switch).   Is the concept correct or is there a better way to do this? Thanks! I can print with printf as a hex or octal number.  Is there a format tag to print as binary, or arbitrary base? I am running gcc. printf(&quot;%d %x %o\n&quot;, 10, 10, 10); //prints &quot;10 A 12\n&quot; print(&quot;%b\n&quot;, 10); // prints &quot;%b\n&quot; Does existence of magnetic monopole break covariant form of Maxwell’s equations for potentials? Absence of magnetic charges is reflected in one of Maxwell's fundamental equations: $$\operatorname{div} \vec B = 0. \tag1$$ This equation allows us to introducte concept of vector potential: $$\vec B = \operatorname{rot} \vec A.$$ Using this concept, it is possible to express Maxwell's equations in a graceful symmetric form: $$\nabla^2 \vec A - \frac{1}{c^2}\frac{\partial^2 \vec A}{\partial t^2} = - \frac{\vec j}{\epsilon_0 c^2}, \tag2$$ $$ \nabla^2 \phi -\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = - \frac{\rho}{\epsilon_0}. \tag3 $$  Noticing, that vector $\vec A$ and scalar $\phi$ potentials, as well as electric current density $\vec j$ and charge density $\rho$, form a 4-vector in Minkovsky space-time. Therefore, Maxwell's equations can be expressed in a covariant form, using d'Alembertian: $$\nabla_{\mu}\nabla^{\mu} A_{\nu} = \frac{j_{\nu}}{\epsilon_0}. \tag4$$  If magnetic monopols exist, Maxwell's equation $(1)$ will look as: $$\operatorname{div} \vec B = \mu_0 c \rho_{\mathrm{magnet}}.$$  As the divergence of $\vec{B}$ isn't equal to zero, it impossible to introduce concept of vector potential. Thus, the equation in the form of $(4)$ will not be possible to express. Different methods of evaluating $\int\sqrt{a^2-x^2}dx$: Is there a simple and nice way to solve $\int\sqrt{a^2-x^2}dx$:  PS:I am not looking for a substitution like $x=a\sin p$, How to write in bibliography (package natbib) letter "a" with two dots above? Specially, I mean the word Birkhäuser.  Is there a general rule or way how to write such umlauts or other accented letters in bibliographies? How do you run your own code alongside Tkinter's event loop? My little brother is just getting into programming, and for his Science Fair project, he's doing a simulation of a flock of birds in the sky. He's gotten most of his code written, and it works nicely, but the birds need to move every moment.  Tkinter, however, hogs the time for its own event loop, and so his code won't run. Doing root.mainloop() runs, runs, and keeps running, and the only thing it runs is the event handlers.  Is there a way to have his code run alongside the mainloop (without multithreading, it's confusing and this should be kept simple), and if so, what is it?  Right now, he came up with an ugly hack, tying his move() function to &lt;b1-motion&gt;, so that as long as he holds the button down and wiggles the mouse, it works. But there's got to be a better way. How is Cooper's age justified? Formula for completing the square? My math teacher said that this was the formula for completing the square.  Original function: $$ax^2 + bx + c$$ Completed square: $$a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$$  However, using this formula I'm not getting the same answers that I would get just by determining the stuff myself.  Is this correct? How should the "and/or" text of a card be interpreted? How should the "and/or" text of a card be interpreted? For example on a card like  the text reads:     Nahiri's Wrath deals damage equal to the total converted mana cost of the discarded cards to each of up to X target creatures and/or planeswalkers.   For the remainder of this question, let's fix X at 2 to keep things simple.  Should the "and/or" be interpreted such that the effect is applied to a combination of targets the player chooses. That is to say for Nahiri's Wrath the player could perform one of the following options:   Nahiri's Wrath deals damage to each of two target creatures. Nahiri's Wrath deals damage to each of two target planeswalkers. Nahiri's Wrath deals damage to each of one target creature and one target planeswalker.   Or should the card's effect be expanded and applied to both sides of the "and/or". With this interpretation the player would have to the following two options when playing Nahiri's Wrath:   Nahiri's Wrath deals damage to each of X target creatures and Nahiri's Wrath deals damage to each of X target planeswalkers Nahiri's Wrath deals damage to each of X target creatures or Nahiri's Wrath deals damage to each of X target planeswalkers   The confusion stems from  (which was formerly formerly worded using the same phrasing) where some commenters have said things like:     So this lets you do a total of 12 damage? 3 each to 2 target creatures AND 3 each to 2 target players, one of which could be a planeswalker? That's how the "and/or" works, right? You can choose. Otherwise it would just be "or"... Find out trace of a given matrix $A$ with entries from $\mathbb{Z}_{227}$ Let $A$ be a $227\times 227$ matrix with entries in $\mathbb{Z}_{227}$ such that all of its eigen values are distinct. What would be its trace? I think it is zero by adding all 227 elements but i am not sure.  Edited: Here I have assumed that eigenvalues are in a base field. I have a couple of illustrations done in Illustrator and I am planing to animate it for a website that I am working on, I've heard of  toolkit with flash, but is it the best way to go or is there another "better" way of doing it?  Here's an example of the kind of animations I am aiming for: