Let $G$ be a graph. Let $k$ be the minimal degree of $G$. Show that $G$ contains a cycle of length $k + 1$.
Let $G$ be a graph of minimum degree $k&gt;1$. Show that $G$ has a cycle of length at least $k+1$.  How would I show this? I understand that a cycle is a sequence of non-repeated vertices and the degree of a graph is the number of neighbors the vertex has.
Reading the target of a .lnk file in Python? I'm trying to read the target file/directory of a shortcut (.lnk) file from Python. Is there a headache-free way to do it? The  is way over my head. I don't mind using Windows-only APIs.  My ultimate goal is to find the "(My) Videos" folder on Windows XP and Vista. On XP, by default, it's at %HOMEPATH%\My Documents\My Videos, and on Vista it's %HOMEPATH%\Videos. However, the user can relocate this folder. In the case, the %HOMEPATH%\Videos folder ceases to exists and is replaced by %HOMEPATH%\Videos.lnk which points to the new "My Videos" folder. And I want its absolute location. Letter of recommendation from Post-Doc or Professor I estimated the sample covariance matrix $C$ of a sample and get a symmetric matrix. With $C$, I would like to create $n$-variate normal distributed r.n. but therefore I need the Cholesky decomposition of $C$. What should I do if $C$ is not positive definite? How well does T-Mobile's (US) Simple Choice plan work when in Europe? How should a GM deal with "standing orders?" Will everything eventually be a single black hole? Given the idea that gravity is always a factor (even though greater distances are excessively tiny), would everything eventually end up forming a single black hole?  My thinking is that, with infinite time, all mass will eventually be pulled together. amsmath align environment: row spacing How can I stretch the spacing between consecutive lines in an align* environment? I have a few lines with \fracs in which both the numerator and the denominator consist of textual phrases, not math symbols, and the result looks too cluttered.  \begin{align*} X &amp;= \frac{\textit{Some text here}}{\textit{And some text here}}\\ Y &amp;= \frac{\textit{Will make everything}}{\textit{Look too close together}} \end{align*} How to model this odd-shaped distribution (almost a reverse-J) How do I install the Tor Browser Bundle in Ubuntu? In Ubuntu 13.04 I used to use the ppa:upubuntu-com/tor64  PPA to install the 64-bit Tor Browser Bundle. However, in Ubuntu 13.10 it doesn't work anymore. How do I install the Tor Browser Bundle in Ubuntu? Is it possible to add subtitles of a figure which contains subfigures I have a single image which contains two sub-images. I want the following subtitles (a) and (b) to be shown correctly.   In which, left sides is my current image which contains two rectangular. And right sides is my expected result. I found a solution is that we can use subfig package. However, it requires separating the original figure into two subfigure. In that question, I am regarding that we do not need split original figure, I want to write (a),(b) directly below figure. Thanks Current solution is  \begin{figure}[H] \captionsetup{labelfont={bf}}   \centering     \subfloat[]{\centering\includegraphics[scale=0.5]{Fig1a.png}} \\     \subfloat[]{\centering\includegraphics[scale=0.5]{Fig1b.png}}   \caption{This is main figure}   \label{fig:9} \end{figure} For any triangle with sides $a$, $b$, $c$, prove the inequality $$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0 .$$   This is IMO 1983 problem 6.  I tried substituting $a=x+y$, $b=y+z$, $c=z+x$ but well it doesn't help in any sense except wasting 3 pages that lead to nothing (please don't mind the joke). Using $a=2R\sin A$, $b=2R\sin B$, $c=2R\sin C$ also didn't lead to anything for me. Could you give me a hint for finding the proper substitution? Frea is trying to kill me I've been trying to play the Dragonborn DLC for Skyrim on the PS3. In order to advance the story, I need to talk to Frea and her dad. But every time I go near the village they try and murder me for slaughtering the village back when I was like level 15 (I'm level 43 now).  I've tried yielding but they won't accept it and I don't know how to make them stop, all I want to do is finish the story. Does anybody know how to fix this issue? Other than looking at the profile, is there a way to find out someone's real name? A user on SO, Abhijit, referred to the OP of a question, acpigeon, as "he" (and then Aaron) .  I didn't see the name on the question page, nor on . Is there a way to look this up that I missed? Alternatives are Abhijit knows acpigeon, or maybe Abhijit was just making things up.  I'm still relatively new to these sites, so I'm still learning my way around. Is there a way short of the two alternatives above to know the OP's name? If so, how? When does $(e^a)^b = e^{ab}$ hold? For a complex number $A$ and a real number $B$, when does the well-known formula $(e^A)^B = e^{AB}$ fail? Or does it hold at all for complex A?     Since $e^{2\pi i} = 1$, if this formula holds for all complex numbers $A$ and real numbers $B$, then it would imply that $e^{2\pi ti}=( e^{2\pi i} )^t = 1^t=1$ for all $t$, which is obviously false. Find all rings $R$ satisfying $\mathbb Z \subset R \subset \mathbb Z[\frac {1}{2}]$ Find all rings $R$ satisfying $\mathbb Z \subset R \subset \mathbb Z[\frac {1}{2}]$.   Solution: Let $f \in R$. Then $f= \frac {m}{2^n}$ for some integers $m,n$.  Since $2^{n-1} \in R$, hence $\frac {m}{2} \in R$. Hence one gets that $ \frac {1}{2} \in R$, so $R=\mathbb Z[\frac {1}{2}]$, so there is no ring $R$ satisfying given conditions.     What are the other possible solutions of this problem?    P.S: You can give a solution using any theorem from commutative algebra also. Is my expansion of the state $| x \rangle$ correct? In my quantum mechanics textbook it says that the relation between the basis $|x\rangle$ and $|p\rangle$ is given by:  $\langle p | x \rangle = \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \, .$  However, i'm not sure how to go about proving this relation. My idea was that the eigenstates of momentum can be written as below:  $\phi(x,p) = \Large \frac{1}{\sqrt{2\pi \hbar}} e^{-ipx/\hbar},$  which form an orthogonal basis.  But, we can expand any state, $|\text{state}\rangle$, in terms of the othorgonal basis right? So is my notation below correct for the expansion of the state $| x \rangle$ ?  $|x\rangle = \int_{-\infty}^{+\infty} \phi(p', x) |p'\rangle dp'$  If it were correct then the following would also be correct?  Using the fact that $\langle p | p' \rangle = \delta(p - p')$, $\langle p | x \rangle$ can be computed:  \begin{align} \langle p | x \rangle =&amp; \int_{-\infty}^{+\infty} \phi(p', x) \langle p | p' \rangle dp' \\ =&amp; \int_{-\infty}^{+\infty} \phi(p', x) \delta(p - p') dp' \\ =&amp; \phi(p, x) \\ =&amp; \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \end{align} as required.  Does the logic work here? Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$ Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $. How does one prove the following limit? $$   \lim_{n \to \infty}   \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots \sqrt{1 + (n - 1) \sqrt{1 + n}}}}} = 3. $$ How to install an application that requires gksu package on ubuntu 18.04? I downloaded the Upwork desktop App on ubuntu 18.04. When I open it with Gdebi package installer I saw this error:    I tried to install gksu manually via sudo apt install gksu but I got this error  Building dependency tree        Reading state information... Done Package gksu is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'gksu' has no installation candidate   How can I install this software on ubuntu 18.04 (I download it yesturday so it is the most recent version on upwork website) ? One single command to update everything in Ubuntu?