Hartshorne's proof of Proposition 2.5, Chapter II of his book Algebraic Geometry
The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$
Why is it that logistic regression becomes unstable when classes are well-separated? What does well-separated classes mean? I would really appreciate if someone can explain with an example. Why is vote time missing in the SE data dump and SEDE? Is there a particular reason why vote times are omitted in the data dump?  IMO this could be a very useful data-set. Is this one of the steps that was done to anonymize the data, or does it simply not exist? This would be a very good addition to future dumps to help build time-based relationships, which are currently very rigid and full of assumptions. some \newcommand instructions not working Is it possible to check for an injured player in Minecraft? I am creating a mini-game in Minecraft, where I must check to see if the player is injured. The injured player will then be given an item. My guess is that the command would look something like /testfor @p[injured=true], but this command doesn't work. Is there a command for this? As Joel points out in , in  (aka: K &amp; R), there is mention of this property of arrays in C: a[5] == 5[a]  Joel says that it's because of pointer arithmetic but I still don't understand. Why does a[5] == 5[a]? I have something like that  how do I convert this to a YouTube playlist? Strange OutOfMemory issue while loading an image to a Bitmap object How to sort an array of integers correctly Where will changes and fixes to guides and manuals for 5e be found? I'm interested in a grappler build, among others, and don't know where to look to see if the last bullet for the grappler feat has been changed. The grappler example aside, it should be useful to know for future fixes. How can I check if a directory exists in a Bash shell script? What command can be used to check if a directory exists or not, within a Bash shell script? Some mesh-edges in object-mode aren't displayed So I have been extruding a cube to make a low poly banana and when I go from edit to object mode half the edges don't appear as well as half is dark.      I tried looking for an solution couldn't find one and its making me feel like I am doing something wrong so I decided to come here. How to place text above the figure/table How to place text above the rotated Figure (LANDSCAPE) in the same page. I tried but text automatically moved to the following page.  TEXT TEXT....   \begin{landscape} \begin{table}[] \centering \includegraphics[width=160mm]{figures2/comparation3.png} \caption{A comparison of Aloha protocols} \label{tab:ComparationAloha} \end{table} \end{landscape}     I added this code, but one part of table disappeared. I must have type TABLE since I need it for list of tables later.  \lipsum[11] \begin{table}[h!] \rotatebox{90}{\parbox{100mm}{ \includegraphics[width=160mm, rotate=90]{figures2/comparation3.png} \caption{A comparison of Aloha protocols} \label{tab:ComparationAloha}}} \end{table} \lipsum[12] What are beginner’s books on typography? Is the Death Star’s gravitational force strong enough to hold an atmosphere? The Death Star is the size of a small moon. Endor being a moon (not sure if it's small or not) has a rich atmosphere.  Could the Death Star hold a nitrogen/oxygen based atmosphere with its gravity?  (Now re-reading my question, I suppose this sounds a lot like the plot of SpaceBalls... that was not my intention) Apache and systemd I've just started using Debian 8 which has introduced systemd. I'm not interested in the politics around SysV vs systemd, but I am confused, especially around Apache (2.4).  There are now two ways to restart apache:   apache2ctl restart or apache2ctl graceful systemctl restart apache2 and it seems systemctl restart apache2.service does the same thing.   And they appear to do different things(!). I don't understand the difference.  I seem to need to do both to get a proper full restart working. e.g. after changing a config file for the php module it only noted the change after both.  Another time when apache2ctl configtest was failing and the failure was reported in systemctl status apache2 I fixed the config so that the first of these commands was happy, restarted with apache2ctl restart but the systemd status command still listed it as not running and with the old config problem.  I promise to spend ages reading about systemd (RTFM) in depth at some point, but meanwhile on a practical level relating to apache, I'd appreciate advice. What is the meaning of this "let there be light" joke? I'm suppose to prove the following...  Let $A$ be a nonempty subset of $\mathbb{R}$. If $\alpha = \sup A $ is finite show that for each $ \epsilon &gt; 0 $ there is an $a \in A $ such that $\alpha - \epsilon &lt; a\leq \alpha $.  This is what I have so far...  Proof: We claim that $\alpha = \sup A $   Then by definition of a supremum $\alpha $ is the least upper bound of set A  Thus $\forall a \in A$,  $a \leq \alpha$   Suppose $\epsilon &gt; 0$   Then $\alpha - \epsilon &lt; \alpha $.   Thus $\alpha - \epsilon &lt; a\leq \alpha$.   I feel like I'm missing some step between my last and second to last step so what am I forgetting? Also is the rest of my proof right? In my question's preview, the attached image looks rotated. So, assuming you are able to see it in the real orientation, can anyone determine what is that dust flake type thing on the top, slightly towards the left?   One looks like a dust flake and there's another dot to the left of it?  Note: I wont have physical access to the camera until the weekend.  Is it just a dust flake on the lens or can it be some problem with the sensor? Its a 3 month old Canon 1500D. The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual proofs on the internet but I was wondering if someone knew a proof that is unexpected in some way. e.g. can you link the theorem to some famous theorem, can you find a non-trivial geometric proof (I can find some of those), proofs that use theory that doesn't link to this inequality at first sight (e.g. differential equations …)?  Induction, backward induction, use of Jensen inequality, swapping terms, use of Lagrange multiplier, proof using thermodynamics (yeah, I know, it's rather some physical argument that this theorem might be true, not really a proof), convexity, … are some of the proofs I know. 80's pop song with lyric "It's the story of my life if you wanna know" I'm trying to identify a song by a female 80's pop group.   Here are the lyrics I remember:     It's the story of my life if you want to know   a play where everything is meant to be   you're the story of my life and I can't let you go