How can I force two figures to appear on the same page, each with their own figure number and caption?
Getting two pictures to appear on the same page
Given a simple graph and its complement, prove that either of them is always connected. I was tasked to prove that when given 2 graphs $G$ and $\bar{G}$ (complement), at least one of them is a always a connected graph.  Well, I always post my attempt at solution, but here I'm totally stuck. I tried to do raw algebraic manipulations with # of components, circuit ranks, etc, but to no avail. So I really hope someone could give me a hint on how to approach this problem. How to calculate $f(x)$ in $f(f(x)) = e^x$? How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$?  Is there a faster-converging method than power series for fractional iteration/functional square roots? Basic buoyancy question: Man in a boat with a stone This comes from a brain teaser but I'm not sure I can solve it:     You are in a rowing boat on a lake. A large heavy rock is also in the boat. You heave the rock overboard. It sinks to the bottom of the lake. What happens to the water level in the lake? Does it rise, fall or stay the same?   I'd say that the water level drops, because when you drop the stone in the lake, the water level rises according to the volume of the stone, BUT the water level decreases by a volume of water weighting the stone's weight, when you take it off the boat.  Is that correct? Is it possible, on Debian with Klipper running, to have a separate "clipboard stack" for a particular application?  I'd like the text that I copy in application Y not to be shared with all the other applications running.  All the solutions mentioned at  seems like overkill. Is it required for panel data to use dummy variables? I am doing research considering seven countries and I have panel data. My question is: do I need to include dummy variables every time I use panel data in regression, or is enough to do it as a fixed effect in Stata, without dummy variables? What determines whether colors you can't see are visible or not? So, when someone is red-green colorblind, the colors appear the same to them, like this:   And if you're totally colorblind, then things presumably just appear like they would in a black-and-white movie.  However, this isn't how ultraviolet patterns seem to work. Compare how we see this flower to the version where ultraviolet is visible:   The UV pattern is completely invisible here. However, unlike with the red and green, this isn't because yellow and UV are colors that appear identical when you can't see UV. Look at these flowers:     This time the flowers are purple, but the UV pattern is still invisible. Why is that? Shouldn't the UV pattern still be apparent on at least one of the flowers, just in a different color? And on some other flowers, the UV does appear as a different color. So:    Why is the UV invisible only sometimes?  Does it have to do with the flower using iridescent structures to produce color, instead of a pigment? Can this happen with red and green, as well? Hash character in LaTeX Possible Duplicate:          How do you write the # character in LaTeX? My computer does not store the history of the entries entered in RUN (WIN+R).   Thus, how to enable Windows to store the history? How to prove that the sum and product of two algebraic numbers is algebraic? Suppose $E/F$ is a field extension and $\alpha, \beta \in E$ are algebraic over $F$. Then it is not too hard to see that when $\alpha$ is nonzero, $1/\alpha$ is also algebraic. If $a_0 + a_1\alpha + \cdots + a_n \alpha^n = 0$, then dividing by $\alpha^{n}$ gives $$a_0\frac{1}{\alpha^n} + a_1\frac{1}{\alpha^{n-1}} + \cdots + a_n = 0.$$     Is there a similar elementary way to show that $\alpha + \beta$ and $\alpha \beta$ are also algebraic (i.e. finding an explicit formula for a polynomial that has $\alpha + \beta$ or $\alpha\beta$ as its root)?    The only proof I know for this fact is the one where you show that $F(\alpha, \beta) / F$ is a finite field extension and thus an algebraic extension. Question about repeating decimal? For simple fraction, we can easily convert it to repeating decimal by calculator. Ex. $\frac 1 3 = 0.33333\ldots$, $\frac 1 7=0.(142857),\ldots$ But some fraction fraction like $10/29, 1/97,...$ The repeating part of them are too long, so it can't fully show on the calculator. So is there any algorithm  to find the repeating part for that fraction? Intuition for Rayleigh PDF We have ground-truth data $\mathbf{x}^* = (0,0)^T \in \mathbb{R}^{2}$. Furthermore, we have $N$ measurements $\mathbf{x}_i \in \mathbb{R}^{2}, i\in \{1,\ldots,N\}$. We calculate the $N$ error vectors $\mathbf{e}_i = \mathbf{x}_i - \mathbf{x}^*$. We assume that the two components of $\mathbf{e}_i = (e_{x,i}, e_{y,i})^T $ are independent of each other and normally distributed with equal mean $\mathbf{\mu} = (0,0)^T$ and variance $\sigma^2$. The absolute errors $e_i = \sqrt {e_{x,i}^2 + {e}_{y,i}^2} $ follow a Rayleigh distribution, is that correct?  The probability density function (PDF) of the Rayleigh function is zero at a value of zero. Why is this the case (intuitively)? I would assume that our errors should most likely be 0 (i.e. PDF should be highest at 0)... Show that $\sum\limits_{n=0}^\infty \frac{z}{(1+z^2)^n}$ converges for all $z\in \mathbb{C}$ outside $|1+z^2|=1$ Show that $\sum\limits_{n=0}^\infty \frac{z}{(1+z^2)^n}$ converges for all $z\in \mathbb{C}$ outside $|1+z^2|=1$  Actually, I have the problem to understand the meaning of "outside $|1+z^2|=1$". If it means that the collection of $z\in\mathbb{C}$ such that $|1+z^2|&gt;1.$ This problem will be easy to verify since for each given $z$ with $|1+z^2|&gt;1$, then we have $|1+z^2|^{-1}&lt;1$. Then, it is the geometric series with common ratio less than $1$ which implies that the series converges.   Do I have correct interpretation with "outside $|1+z^2|=1$"? If not, what is the correct interpretation with this? Thanks. Near-universally applicable alternative to "sanity check" The phrase "sanity check" is  as it implies that there is something wrong with people who have mental illnesses and the word "sanity" has been used to discriminate against such people.  Therefore, it should should be avoided.  What is a better alternative in the context of software?  Example usage:     _: if this is not what it should be, we have memory corruption (a fatal   error) and should terminate the program. How to generate all permutations of a list? Why must $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ -- the set of all polynomials in $\sqrt{2}$ and $\sqrt{3}$ with rational coefficients  -- contain multiplicative inverses?   I have gathered that every element of $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ takes form $a + b\sqrt{2} + c\sqrt{3} + d\sqrt{2}\sqrt{3}$ for some $a,b,c,d \in \mathbb{Q}$, and I have shown that $\mathbb{Q}[\sqrt{2}]$ and $\mathbb{Q}[\sqrt{3}]$ are both fields, but it's not clear to me why this allows for general multiplicative inverses to exist in $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$. I want to kill a whole process tree.  What is the best way to do this using any common scripting languages? I am looking for a simple solution. I am wondering how to recreate some kinds of effects (you can easily do in Photoshop) in Blender.  I want to create the gradient textures like ones pictured below, should I render objects and blur them in compositor? Or can I achieve it using only material nodes? Securing Cable when installing old work box I am installing an electrical outlet above a gas fireplace for TV.  I understand that electrical code requires securing the NM cable within 12" of outlet box.  I am curious how one goes about achieving this in a finished wall being that there is really no room to get hand, hammer, drill or screwdriver inside the wall through that box opening.   I am sure this is straightforward for experienced electricians but I have no idea how to go about this. Nested List Indices How do I Uninstall Windows 10 or Windows 10 preview?