I have a question on a comment from Daniel Hyubrechts' Complex Geometry  on pages 133/134.  Let $X$ be a compact Kähler manifold. Consider the exponential sequence on cohomology   $$ ... \to H^1(X,\mathcal{O}_X^*)=Pic(X) \to H^2(X, \mathbb{Z}) \to H^2(X, \mathcal{O}_X) \to ...$$     Remark 3.3.3 Often, the image of $Pic(X) \to H^2(X,\mathbb{R}) \subset H^2(X,\mathbb{C})$ is called the Neron-Severi group $NS(X)$ of the manifold $X$. It spans a finite dimensional real vector space $NS(X)_{\mathbb{R}}= NS(X) \otimes \mathbb{R} \subset  H^2(X,\mathbb{R}) \cap H^{1,1}(X)$, where the inclusion is strict in general. The Lefschetz theorem above thus says that the   natural inclusion $NS(X) \subset H^{1,1}(X, \mathbb{Z})$ is an equality.      If $X$ is projective, yet another description of the Neron-Severi group can be   given. Then, $NS(X)$ is the quotient of $Pic(X)$ by the subgroup of numerically   trivial line bundles. A line bundle $L$ is called numerically trivial if $L$ is of   degree zero on any curve $C \subset X$.    Q1: Why in case of $X$ projective i.e. closed subscheme of projective space, these two descriptions of Neron-Severi group coincide? Thus why the image of $Pic(X) \to H^2(X,\mathbb{R})\subset H^2(X,\mathbb{C})$ coincides with quotient $Pic(X)/A$ with $A$ subgroup of numerically trivial line bundles?  Q2: Why $NS(X)_{\mathbb{R}} $ is contained in $  H^{1,1}(X)$. Recall, we have decomposition $H^2(X, \mathbb{C})=H^{2,0}(X) \oplus H^{1,1}(X) \oplus H^{0,2}(X)$.
Let $X$ be a smooth projective variety over $\mathbb{C}$, then the Neron-Severi group $NS(X)$ of $X$ is defined to be the Picard group of $X$ modulo algebraically equivalent relations.   On the other hand, by the exponential sequence, there is a first Chern class map  $$c_1: {\rm Pic}(X) \to H^2(X, \mathbb{Z}).$$ It is claimed that the image of $c_1$ coincides with $NS(X)$.   I want to know why this is true. Any suggestion or reference is greatly welcome!
Why doesn't my systemd user unit start at boot? Which characters are valid in CSS class names/selectors? What characters/symbols are allowed within the CSS class selectors? I know that the following characters are invalid, but what characters are valid?  ~ ! @ $ % ^ &amp; * ( ) + = , . / ' ; : " ? &gt; &lt; [ ] \ { } | ` # Can I store the .git folder outside the files I want tracked? I have an unusual idea to use git as a backup system. So let's say I have a directory ./backup/myfiles and I want to back that up using git. To keep things clean I don't want to have a .git directory in the myfiles folder, so I thought I could create ./backup/git_repos/myfiles. From looking at the git docs, I've tried doing this:  $ cd backup/myfiles $ mkdir ../git_repos/myfiles $ git --git-dir=../git_repos/myfiles init Initialized empty Git repository in backup/git_repos/myfiles/ $ git --git-dir="../git_repos/myfiles/" add foo fatal: pathspec 'foo' did not match any files   You can see the error message I get there. What am I doing wrong? A friend of mine accidentally left a carton of eggs on her counter, unrefrigerated, for three days. The eggs had been previously refrigerated both at the store and at home. Now she's planning to do some more cooking which requires eggs, and is wondering if it's still safe to use them for baking. I believe she is planning on baking cookies with them, so they would be baked at fairly high temperatures for probably at least 10 minutes. Would this be safe, or are eggs left unrefrigerated for that long not safe for consumption? Determinant of block matrix with commuting blocks I know that given a $2N \times 2N$ block matrix with $N \times N$ blocks like   $$\mathbf{S} = \begin{pmatrix} A &amp; B\\  C &amp; D \end{pmatrix}$$  we can calculate $$\det(\mathbf{S})=\det(AD-BD^{-1}CD)$$ and so clearly if $D$ and $C$ commute this reduces to $\det(AD-BC)$, which is a very nice property.   My question is, for a general $nN\times nN$ matrix with $N\times N$ blocks where all of the blocks commute with each other, can we find the determinant in a similar way? That is, by first finding the determinant treating the blocks like scalars and then taking the determinant of the resulting $N\times N$ matrix.  Thanks in advance! C non-blocking keyboard input I'm trying to write a program in C (on Linux) that loops until the user presses a key, but shouldn't require a keypress to continue each loop.  Is there a simple way to do this? I figure I could possibly do it with select() but that seems like a lot of work.  Alternatively, is there a way to catch a ctrl-c keypress to do cleanup before the program closes instead of non-blocking io? How do I redirect to another webpage? From time to time I see an enum like the following:  [Flags] public enum Options  {     None    = 0,     Option1 = 1,     Option2 = 2,     Option3 = 4,     Option4 = 8 }   I don't understand what exactly the [Flags] attribute does.  Anyone have a good explanation or example they could post? What are neutral density filters and how do I use them to create long exposures in daylight? I've seen a few related posts on the site but nothing specifically on this point:  I'd like to be able to take long exposure photos during daylight (like the examples in ). I'm told that you can do this by using a neutral density filter. So my question is: what is a neutral density filter exactly, and how is it used to achieve that smooth long exposure effect? Question is :  Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, Prove that  $Y\cup A$ and $Y\cup B$ are connected.  What i have tried is :   Suppose $Y\cup A$ has  separation say $C\cup D$ then all i could see is that   Either $Y\subset C$ or $Y\subset D$ as $Y$ is connected and $ C\cup D$ is separation for a subset that contains $Y$.  Without loss of generality we could assume $Y\subset C$  As $Y\cup A=C\cup D$ and $Y\subset C$  i can say $D\subset A$ (I do not know how does this help)  I have $X-Y=A\cup B$ with $A\cap B=\emptyset$  I have not used connectedness of $X$ till now, So i thought of using that and end up with following :   $X-Y=A\cup B\Rightarrow X=Y\cup A\cup B=(Y\cup A)\cup (Y\cup B)$  I do not know what to conclude from this...   I would be thankful if some one can help me by giving some "hints"  Thank you Storage order vs Result order Is it possible to set the default size of the Empty Object? The problem is since I switched my blender scale to metric, the Empty's default size is kinda large for most times I need it. curlicue letters with mathptmx package How to show letters like  with mathptmx package?  I'm using pdfLaTeX with mathptmx package becasue its gooding looking normal math font letters a-z, A-Z. However, if only with this package, \mathsrc{...} is reported as "undefiend sequence" and \mathcal{...} is curlicue style like the following:  \mathcal{F} shows . If I add \usepackage{mathrsfs}, \mathcal{F} and \mathscr{F} are the same as .  What I expect is \mathcal{F} shows  and \mathscr{F} show the one in previous line.   What should I adjust? I hope mathptmx package is still being used.  Minimal code script goes like that:  % using PdfLaTeX \documentclass[journal]{IEEEtran} \usepackage{amssymb,amsmath,amsfonts} \usepackage{mathptmx}  \usepackage{mathrsfs} \begin{document} $$\mathcal{F}$$ $$\mathscr{F}$$ they are the same \end{document} You can also contact me to customize your template, or create a new unique one.   Do I need to put a comma between the new and unique adjectives in this sentence?  In my native language (if I am not mistaken) we don't put a comma if the adjectives are of a different type. Not sure about English though, and if it's the case here anyway. Any help would be appreciated! I know of 2 basic kinds of default web icons for websites:   : place an .ico formatted file at the root of your site named favicon.ico and most desktop browsers will find it and display it in various contexts (bookmarks, tabs). Can also be added in code, which allows you to specify other names and other graphics types to the file, though there are :  &lt;link rel="shortcut icon" href="/favicon.ico" /&gt; &lt;link rel="icon" href="/favicon.ico" /&gt;  : place a png formatted file called either apple-touch-icon.png or apple-touch-icon-precomposed.png at the root of your site and iPhone OS/iOS versions of Safari will pick those up and store them when the site is added to the Home Screen. Can also be added with code like:  &lt;link rel="apple-touch-icon" href="/custom_icon.png" /&gt;    Are there any other default icon graphics for a website a webmaster can include? 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] I am trying to install Ubuntu 12.10 for quite some time, and passing hurdles one by one. Now I am in a situation as follows.  I have got a PC and 10 GB HDD which will be totally dedicated to Ubuntu so no option of Wubi and dual boot.  I was trying to install from DVD, but it is getting stuck at "Out of frequency" error. So I had to adapt for USB boot option. But my PC is USB non bootable, so workaround is "Plop Boot Manager". So I am doing the installation procedure as follows:   starting  from a CD drive which is having plop installed. opting for for USB boot in plop options. booting begins from USB. monitor eventually gives "out of frequency" error press Shift+Alt+F1 to get the terminal. open the grub with sudo nano /etc/default/grub. do necessary changes. sudo update-grub.   Now here I am getting error as follows:   /usr/sbin/grub-probe:error:failed to get canonical path of /cow.     My system is   P4 3.06 GHz, 1 GB ram , 10 GB HDD without an OS, monitor CRT lg StudioWorks (7 years old). Mobo Mercury P4 266a NDMx (865 equivalent). The whole system is perfectly in working condition under XP, but it is USB non bootable, and all other devices working perfectly.  What should I do next? Two similar limit problems I have two limit problems which are quite similar, so I've put them both into this post.   $((4^{10}+2^{n})^{\frac{1}{n}})$ $((3n^{2}+n)^{\frac{1}{n}})$   Attempt:  For 1, I'm not sure if this is permissible. I know that $\lim_{n \to \infty}(x^{n}+y^{n})^{\frac{1}{n}} = \max\{x,y\}$. Obviously $4^{10}$ is a constant so it's not quite in the same form, but I believe it is still okay to use this result, and so the limit is $2$.  For 2, I've got $((3n^{2}+n)^{\frac{1}{n}}) = (n(3n+1)^{\frac{1}{n}})$, but then I'm not too sure how to proceed.   I also thought of taking the log and proceeding that way, however in the notes I'm learning from they haven't covered l'hopital yet, so I'm trying to not use it.  Thanks. I came across a user with a notification that states:     This account is temporarily suspended network-wide. The suspension period ends on Dec 2 '18 at 2:35.   Screencap for posterity:    Unless this user is  (in which case the full year should be shown, or at least a tooltip), it seems like they should have been unbanned 4 months ago. In how many ways can $7$ girls and $3$ boys sit on a bench in such a way that every boy sits next to at least one girl In how many ways can $7$ girls and $3$ boys sit on a bench in such a way that every boy sits next to at least one girl  The answer is supposedly $1,693,440 + 423,360 = 2,116,800$