17.676 proof-verification questions.

1 How many colorings are in a complete bipartite graph which is planar and has Eurlerian path?

Let $G$ be a complete bipartite graph of $7$ vertices. $G$ is also planar and has Eulerian path. What is $G$'s chromatic number? I think that because $G$ has Eulerian path then it must have two ...

Second-order differential equation with initial conditions

I try to solve using laplace transform $y'' + y = \sin (t)$ with $y(0) = 1$ and $y'(0) = 2$ but I don't get a solution, I don't know why. I check my work and it seems fine. My calculations Is ...

1 Finding all $\beta \in E$ such that $\beta^2 \in F$. Is it really this easy?

1 answers, 19 views proof-verification field-theory
My friend's assignment has the following question: Let $F$ be a field and $E=F(\alpha)$, where $\alpha\notin F$ and $\alpha^2 \in F$. Find the set of all $\beta \in E$ such that $\beta^2 \in F$. ...

Cantor set is compact?

Cantor set See the link, I am referring to cantor set on real line. I wish to show that it is compact. I am doing this buy pointing following arguments. I am not sure if this is enough. Cantor set ...

1 Prove that if f is differentiable on $(a, b)$ and $f'$ is increasing, then $f$ is convex on $[a, b]$

A function $f$ is called convex on an interval $[a, b]$ if, for any $x, y \in [a, b]$ and $t \in [0, 1]$ we have $f(tx + (1 − t)y) \leq tf(x) + (1 − t)f(y)$. Would drawing a picture of this help in ...

$(ab,ac)=a(b,c) , a >0$

$(a,b)$ denotes G.C.D of $a$ and $b$ Let $(b,c)=d$. So, $d|b$ and $d|c$. Claim : $(ab,ac)=ad$. As d|b and d|c , so $d|b.b...b$ ($a$ times) which means $d|ab$. similarly $d|ac$. So d is common ...

3 Probability of flopping a royal flush

I have no stats training, so I am asking if I am attacking this simple statistical problem correctly. What are the chances of flopping a royal flush in Texas hold’em? My attempt: The first card ...

Determine where a function is local injective

I'm studying Multivariate Calculus and I've just studied the inverse theorem and now I'm doing some exercises. There are some questions about local injectivity that are causing some doubts. For ...