Let $A, S$ be commutative rings with identity, and assume $S$ is a finitely generated $\mathbb{Z}^{\geq 0}$-graded $A$-algebra. If $f\in S$ is a homogeneous element of positive degree, $S_f$ is a $\mathbb{Z}$-graded $A$-algebra. Is $(S_f)_0$ (the ring of degree $0$ elements) finitely generated as an $A$-algebra?     It seems to come down to a simple combinatorics problem with exponents that i'm too dumb to do.   (why?: I'm trying to show quasiprojective $A$-schemes have finite type)
Let $S = \bigoplus_{n\ge 0} S_n$ be a graded commutative ring. Let $f$ be a homogeneous element of $S$ of degree $&gt; 0$. Let $S_{(f)}$ be the degree $0$ part of the graded ring $S_f$, where $S_f$ is the localization with respect to the multiplicative set $\{1, f, f^2,\dots\}$. Suppose $S$ is finitely generated algebra over $S_0$. Then $S_{(f)}$ is a finitely generated algebra over $S_0$.
Wrapfigure in an enumerate environment. I want to use wrapfigure in the enumerate environment. For example,   \documentclass[11pt, a4paper]{article} \usepackage{graphicx} \usepackage{float} \usepackage{wrapfig}  \begin{document} \normalsize {\bf Multiple Choice: } \begin{enumerate}     \item \begin{wrapfigure}{r}{4.5cm}           \includegraphics[scale=0.5]{Exam1_Fig1.jpg}           \end{wrapfigure}           As shown in the figure. This rock is phaneritic and contains quartz, K-feldspar, and plagioclase in nearly equal amounts. What is it? \\           (A) rhyolite \\           (B) basalt \\           (C) diorite \\           (D) ash-flow tuff \\           (E) granite \\ \end{enumerate} \end{document}   However, the figure inserted is not wrapped by the text, but put in a strange position. How to fix the problem?   Thank you! How do I turn off equation auto numbering I have two Ubuntu Software Centers, one can be opened, another one can't How to merge vertices in Blender? I am attempting to create my first 3D character just for practice. Being one of my first 3d models I going through the learning curve needless to say. In doing so I find myself creating many extra vertices. Where we will say 10 vertices on a rounded shape are spaced out on the equator and on the other north pole end of the edges they are very close. The close end could all be one vertex. Is it possible to say merge all points to one point and delete the rest? Show that $a \star b=a \cdot b+a+b$ is binary operation for the group $\Bbb{ Q} \setminus \{-1\}$ The group $\left(\Bbb{ Q}\setminus\{-1\},\star\right)$ has as its underlying set the rational numbers different from $-1$ and the operation $\star$ is defined as $a \star b=a \cdot b+a+b$ where multiplication/addition are the usual operations with rational numbers.Show that this is a binary operation.  What I did (Please help verify):  identity element has to be 0 since $a\star 0 = a+0+a(0)$ so for $a\star b=0$ we have  $ab+a+b=0$ $$ab+a=-b \\ a(b+1)=-b \\ a=\frac{-b}{b+1}$$ so $b$ cant be $-1$ since that solution is not in $\Bbb{Q}$. Similar question might have been answered here. Is this operation true for all groups? I think so.  Thanks for the tips and help. Full page image as title page Background  Wrote a technical manual in LyX. The technical manual currently has two parts:   Cover image (PDF) Manual text (PDF)   Problem  When merging the two PDF files (with GhostScript or other software tools), there are a number of issues:   Bookmarks (lost in merging) Colours (obvious changes) PDF tags (lost in merging)   I would rather include the cover page image as a full page (no margins) at the start of the manual and have LyX (LaTeX) generate the front page when it creates the PDF. In this fashion, the cover page and the manual text need not be merged as a separate step.  Question  How do you instruct LaTeX to generate a full page image as the first page in a PDF using KOMA-Script?  Update  Using the pdfpages package produces the following result:    Note the white space above and below the image. The white space is removed using the following:  \includepdf[fitpaper]{cover}   Update  Part of the preamble contains:  % Resize figures that are too wide for the page. \let\oldincludegraphics\includegraphics \renewcommand\includegraphics[2][]{%   \graphicsformat{%     \oldincludegraphics[width=\ScaleIfNeeded,#1]{#2}%   }% }   The full preamble is at:   The error message is:  !pdfTeX error: pdflatex (file cover.pdf): cannot find image file   The file ch_02.lyx and test.latex are in the same directory. The file test.latex is the document in Martin's post. The file test.latex can find the file cover.pdf, yet the file ch_02.lyx cannot find the file. Both use the following code:  \includepdf[fitpaper]{cover.pdf}   Update  The most minimal LyX example that produces the problem:  #LyX 1.6.7 created this file. For more info see http://www.lyx.org/ \lyxformat 345 \begin_document \begin_header \textclass scrbook \begin_preamble \usepackage{pdfpages} \end_preamble \use_default_options false \language english \inputencoding utf8 \font_roman lmodern \font_sans helvet \font_typewriter courier \font_default_family default \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100  \graphics default \float_placement H \paperfontsize default \spacing single \use_hyperref true \pdf_bookmarks true \pdf_bookmarksnumbered false \pdf_bookmarksopen false \pdf_bookmarksopenlevel 1 \pdf_breaklinks false \pdf_pdfborder false \pdf_colorlinks false \pdf_backref false \pdf_pdfusetitle true \papersize letterpaper \use_geometry false \use_amsmath 2 \use_esint 1 \cite_engine basic \use_bibtopic false \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \bullet 0 5 11 -1 \bullet 1 5 24 -1 \bullet 2 0 0 -1 \tracking_changes false \output_changes false \author ""  \author ""  \end_header  \begin_body  \begin_layout Standard \begin_inset ERT status collapsed  \begin_layout Plain Layout   \backslash includepdf[fitpaper]{cover.pdf} \end_layout  \end_inset   \end_layout  \begin_layout Standard Test \end_layout  \end_body \end_document   Thank you! I tried to use the absolute convergence test. $$\int_1^2\left|\frac{\sin{x}}{\log{x}}\right| \, dx&lt;\int_1^2\left|\frac{1}{\log{x}}\right| \, dx$$ but I'm not sure whether $\frac{1}{\log{x}}$ converges or not.  Thanks in advance! At what point does a caster define the target of a spell? The new School of Invention tradition for Wizards () while still in playtesting (and indeed a bit silly in my opinion, more of a "chaos wizard"), got me thinking about its reckless casting ability. This allows the caster to cast magic they do not have prepared and get a random result. The random spells have different targeting criteria (Wall of Stone, Fireball, Feign Death , Mass Cure Wounds) and so it is not possible to decide the target of the spell until it has been cast, as you don't know what you are going to get until it is cast.  This suggests that the designer of the subclass has made the assumption that you don't actually target until you have cast the spell and release it, which makes sense from one point of view, strongly supported by the way a prepared action casting works, PHB p.193 where you cast the spell and hold the energy until the condition is met for the readied action at which point you target the spell.  While the Unearthed Arcana is not RAW or finalized and released, my question is not about this subclass, but about the official rules as a whole.  Is there was any direct RAW, or other supporting or opposing RAW, that states at what point a caster defines the target of a spell?  As a sub-question: are there any circumstances where it actually matters, other than the already defined Readied Action spell casting, in the official rules? Converting LiDAR files from LAZ to LAS format I'm trying to convert LiDAR files from the LAZ format (the compressed version) into LAS (the uncompressed version) using  software.   However, I am not really sure how it works. I created a new output folder as I entered in the app interface, but it is totally empty.  I also had tried the following code in the command prompt window:   D:\LiDAR\Laszip\laszip.exe *.laz    and got a message saying 'no input specified'.  In my folder, there are about 100-ish.laz files as well as one imu.laz. Any suggestions? When in a group chat, if many messages are exchanged between other people in the group, when I check my Whatsapp, the messages are grouped according to sender and not chronologically. I know that my phone isn't the problem as I have recently changed my phone, however, I'm still experiencing the same problem.  I have tried uninstalling and re-installing WhatsApp on my phone, and the same problem still occurs. How to order the Ubuntu Phone? Yes, of course, I want that Ubuntu Phone.   Where is the order now button? How to represent it with UNIQUE indexes and foreign keys? I'm trying to run an interactive command through paramiko.  The cmd execution tries to prompt for a password but I do not know how to supply the password through paramiko's exec_command and the execution hangs.  Is there a way to send values to the terminal if a cmd execution expects input interactively?  ssh = paramiko.SSHClient() ssh.connect(server, username=username, password=password) ssh_stdin, ssh_stdout, ssh_stderr = ssh.exec_command("psql -U factory -d factory -f /tmp/data.sql")   Does anyone know how this can addressed? Thank you. product distribution of two uniform distribution, what about 3 or more Proving Binomial Identity without calculus How to establish the following identities without the help of calculus:  For positive integer $n, $ $$\sum_{1\le r\le n}\frac{(-1)^{r-1}\binom nr}r=\sum_{1\le r\le n}\frac1r $$  and  $$\sum_{0\le r\le n}\frac{(-1)^r\binom nr}{4r+1}=\frac{4^n\cdot  n!}{1\cdot5\cdot9\cdots(4n+1)}$$ What's the significance of conditional and absolute convergence? I know how to find if a series is conditional or absolute convergence by taking the absolute value and seeing if that absolute sum converges (then the original sum is absolute). If the absolute diverges but the original converges, then that's conditional.   But my question is: is there a significance? Is absolute convergence series bigger than absolute? Does a series being absolute or conditional affect rate of convergence, or radius? Does it affect symmetry?   Looking at Google, I only found tests on how to find if it's absolute or conditional. But I want to find meaning or significance. Or at least more mathematical properties. Creating and configuring .bash_profile I was setting path for anaconda. an answer to a question  has the following solution. I searched for .bash_profile, but I can't seem to find it. How do I setup this file and get it to run on startup? Just wanted some feedback to ensure I did not make any mistakes with this proof. Thanks!  Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$, and $|G| &gt; 1$, so for some $x\in G$ we have $x\neq 1$ and $\langle x\rangle\leq G$, so $\langle x\rangle = G$. Suppose $x$ has either infinite order or even order. Then $\langle x^2\rangle \leq G$ but $\langle x^2\rangle \neq \langle x \rangle$, a contradiction. So $x$, and therefore $G$, has odd order, and by Feit-Thompson, $G\cong Z_p$ for some prime $p$.  Edit: Thanks, I see that Feit-Thompson is too much.  Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$, and $|G| &gt; 1$, so for some $x\in G$ we have $x\neq 1$ and $\langle x\rangle\leq G$, so $\langle x\rangle = G$. Suppose $x$ has infinite order. Then $\langle x^2\rangle \leq G$ but $\langle x^2\rangle \neq \langle x \rangle$, a contradiction. So $x$, and therefore $G$, has finite order. Suppose $x$ has composite order $n$ so for some $p &gt; 1$ that divides $n$, $\langle x^p \rangle$ is a proper non-trivial subgroup of $G$, so $G$ is not simple. So $G$ is a cyclic group of prime order. Workaround for Android 4.4 not being able to write to SD cards? What is the scope of variables in JavaScript?