Let $\mathbb{P}^n$ be the complex projective $n$-space and let $\mathcal{H}$ be the dual bundle of the tautologic line bundle over $ \mathbb{P}^n$. I have a feeling that the author of a paper I'm reading is calling a section of  $\mathcal{H}^{\otimes r}$ a &quot;homogeneous polynomial of degree $r$ in $n+1$ variables&quot;. I can (kind of) roughly see why one might conceive of a section of this bundle as such. My question is: is there an exact correspondence between homogeneous polynomials of degree $r$ in $n+1$ variables and sections of $\mathcal{H}^{\otimes r}$, or is it just convenient and intuitive shorthand, or neither?
In chapter 2 section 7 (pg 151) of Hartshorne's algebraic geometry there is an example given that talks about automorphisms of $\mathbb{P}_k^n$. In that example Hartshorne states that $\mathcal{O}(-1)$ has no global sections. However, we know that $\mathcal{O}(1)$ is generated by global sections. This is stated at the first of the section that if $\mathbb{P}_k^n = Proj k[x_0,...,x_n]$ then the $x_0,...,x_n$ give rise to global sections $x_0,...,x_n\in\Gamma(\mathbb{P}_k^n,\mathcal{O}(1))$.  I guess I don't understand this twisted structure sheaf very well, or to be honest I don't think I understand structure sheaves in general as well as I would like. The twisting part seems simple at first- you shift the grading of Proj over and then take the structure sheaf.  Admittedly I don't feel comfortable using the structure sheaf other than using the basic facts about it that Hartshorne gives when it is introduced. If anyone could give some insight as to what's going on here or how I might be able to understand this better it would be much appreciated. Thanks.
Confirm and agree to negative questions I've been racking my brain trying to think of a grammatically correct sentence that uses same without the earlier in the same (see!) phrase.  It is the same It is the very same I have 10 things all the same I have 10 of the same things The same boy as last time did it again The boy did the same thing as last time  I am discounting same-day dry cleaning and any similar phrases that might crop up because using a hyphen is acceptable which suggests it's an atomic phrase and it sounds awkward if inverted:  They have a same-day dry cleaning service. Their dry cleaning service is same-day. - Sounds odd but might be acceptable in some situations Their dry cleaning service is done on the same day. - Sounds better  So I don't think it's possible when same is used by itself. Can someone either explain why same must always have the before it or provide a grammatically correct counter-example? What are the numbers / symbols to the right of my reputation counter? Introduction to proofs with a fair amount of hand-holding? TikZ externalization with absolute path and MiKTeX distribution This code:  \documentclass[11pt,openright]{book} \usepackage[T1]{fontenc} \usepackage[latin9]{inputenc} \usepackage[british]{babel}  \usepackage{tikz} \usetikzlibrary{external} \tikzexternalize[prefix={"C:/Users/MyName MySurname/Documents/LaTeX/Figures/"}]  %\tikzexternalize[prefix={Figures/}]   \begin{document} \begin{tikzpicture}     \draw (0,0) circle (1cm); \end{tikzpicture} \end{document}   gives me the error:     ! I can't write on file `"C:/Users/MyName   MySurname/Documents/LaTeX/Figures/MyMWE-figure0.md5"'.   I'm working with Windows 10, MiKTeX 2.9 distibution.  Following the advice Tarass gave me on a comment of this question:  and as indicated here:  I've registered a user-managed TEXMF directory but I still get the same error.  With \tikzexternalize[prefix={Figures/}] all works perfectly but, since I'm using LyX, the files are stored in a LyX temp dir, hence I loose them when I close the program (making externalization useless), so I need to use an absolute path. I seem to have screwed up something that allows you to inset evenly, I think I have tried every combination of settings in inset and pivot center and even is far from even.  I tried uninstalling and re installing blender 2.71 and get the same result.  The image attached is a simple default cube that I tried to inset evenly.  I have also attached the blend file link  help?? Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$ How to open a serial port by friendly name? How to add features to a PostGIS layer using Quantum GIS? I am using Quantum GIS and the PostGIS dbms. I am able to call the PostGIS layer into QGIS but i am unable to add feature to it. I am using the method of adding features to a Shapefile but still it is not getting reflected in the layer, and I don't know how to save it to the PostGIS database if I am able to add the feature to the PostGIS layer.  I may not have used some technical term for this problem because i don't know what to call it, I am expressing the problem in my own terms. how to create a roster sheet in sharepoint 2010 This is about training management, when an employee requests for training, it goes to approval workflow and once approved, it is like the employee is enrolled.  Once it is approved I want a roster sheet to be created for that course for all the employees registered and who got approved.   The report I need to generate has to be of the enrolled users of one course and one particular session  If I give copy listitem as approved then different courses would come together in the list  The format for the report What is the word for the order of succession in which son assumes the throne after father? It can be any son-younger son after elder son after father. Also, what can be the term for only a single family in power? Please answer. So I was writing up a simple binary tree in Python and came across [...]  I don't believe this to be related to the Ellipsis object, more it seems to have something to do with an infinity loop (due to Python's shallow copy?). The source of this infinity loop and why it doesn't get expanded while expanding when accessed is something I'm completely lost to, however   &gt;&gt;&gt; a [[[[[], [], 8, 3], [[], [], 3, 2], 6, 3], [], 1, 4], [[], [], -4, 2], 0, 0] &gt;&gt;&gt; Keys(a)#With a+b [0, 1, 6, 8, 3, -4] &gt;&gt;&gt; Keys(a)#With [a,b] [8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...]] &gt;&gt;&gt; Keys(a)[1]#?? [8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...], 8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...]]     Version using a+b  def Keys(x,y=[]):     if len(x):y+=[x[2]]+Keys(x[0],y)+Keys(x[1],y)#Though it seems I was using y=y[:]+, this actually outputs an ugly mess     return y   version using [a,b]  def Keys(x,y=[]):     if len(x):y+=[x[2],Keys(x[0],y),Keys(x[1],y)]     return y   So what exactly is [...]? I noticed that there is a definition of the pvalue in my textbook is defined says the following:    I have no idea why it is written with a supremum. I've spent hours pondering this, does anyone have enough of a background to help me with this? Thank you! How to understand the Ubuntu file system layout? JavaScript variables declare outside or inside loop? Exporting Data From Salesforce - REST or SOAP? We have a requirement where we need to export records from salesforce tables into our application programmatically. I am reading on the various API's for data integration from salesforce mainly SOAP vs REST API's.  From what I understand, REST should be used when the web service requests are from mobile devices or browser based clients (since they understand JSON). Is that true? If that's the case how do Bulk REST API's fit the bill (since they are meant for bulk data transfers)?  I would like to get your thoughts on what are pros and cons in using SOAP or REST API's to export data records. Note that the requests are being made a java ee server application programmatically. After retrieving the records (probably in batches) from salesforce records we process (validate, filter etc) them and load them in our internal tables.  Thanks. Why doesn't a \foreach variable work in \draw in an \axis? I'm finding some weird behavior with TikZ's \foreach within an \begin{axis} environment.  This works:  \foreach \x in {-2,-1,...,2} {   \addplot[thick, domain=0.05+(\x*2-1)*pi/2:+(\x*2+1)*pi/2-0.05] (x, {tan(deg(x))});  }   but this doesn't:  \foreach \x in {-2,-1,...,2} {   \draw (\x,-5) -- (\x,5); }   It gives the error  ! Undefined control sequence. &lt;argument&gt; \x                ,-5 l.771     \end{axis}   Here's the really weird part: it only fails when the axis has defined bounds. For example, here's a MWE that works (but doesn't draw anything):  \documentclass{standalone} \usepackage{pgfplots} \begin{document} \begin{tikzpicture}   \begin{axis} % [ ymin=-3, ymax=3, xmin=-5, xmax=5]     \foreach \q in {-2,-1,...,2} {       \draw (axis cs: \q,-5) -- (axis cs: \q,5);     }   \end{axis} \end{tikzpicture} \end{document}   but removing the comment causes it to fail.  Why is this and how can I prevent it?  (My end goal is to draw five segments of the tangent function and their verical symptotes.) can't find using System.Web.UI.HtmlControls namespace Find the image of (1,1,1) by T Question: Find the image of (1,1,1) by T  $$T(x,y,z) = (y,6y,x+y+z)$$   I found the standard matrix to be:   \begin{bmatrix}0&amp;1&amp;0\\0&amp;6&amp;0\\1&amp;1&amp;1\end{bmatrix}  I'm confused on how to approach this question.   In order to find the image is it possible to inverse the values of it's standard matrix?  Would I able to use the identity matrix of 3 and inverse the values of that to find the image?    Any help will be greatly appreciated. Is it possible to embed a scrollable frame inside a PDF? (This is an end-of-year crazy question)   Is it possible to embed a (scrollable) frame or a PDF inside a PDF? If so, how to do it in (pdf)LaTeX?  This is something pretty common in HTML (see for example ). Here it is a webpage with PDF frames   Here it is a fake screenshot so you get the idea.  Portable and non-portable (e.g. only-Acrobat) answers are welcomed.     Example of a down arrow below a frame. The up arrow can be on top of the frame and the triangles point up.