Suppose that we have a commutative ring $R$ and an exact sequence of graded R-modules $$0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$$ It is true that we have an exact sequence of graded R-modules $$0\rightarrow\text{S}^{\bullet}(M')\rightarrow\text{S}^{\bullet}(M)\rightarrow\text{S}^{\bullet}(M'')\rightarrow 0$$ where $\text{S}^{\bullet}(-)$ denotes the symmetric algebra? Thank you for your time.
Let  $$0 \rightarrow M \rightarrow N \rightarrow P \rightarrow 0$$  be an exact sequence of $A$-modules, where $A$ is a commutative ring with unity.  Then for each natural number $i$ one can look at the induced maps  $$\mathrm{Sym}^i(N) \rightarrow \mathrm{Sym}^i(P)$$  and   $$\mathrm{Sym}^i(M) \rightarrow \mathrm{Sym}^i(N).$$  The first question is, if they are still surjective, resp. injective, i.e. is $\mathrm{Sym}^i$ an exact functor?  The second question is: what are the kernel, resp. cokernel of these maps?
I know that $\operatorname{Spec} \mathbb{C}[x]$ can be identified with the set $\mathbb{C}\cup *$, where $*$ is a generic point via the correspondence  $$ \prod_{i}(x-a_i) \leftrightarrow \{a_i\}_i , \ \ \ \ (0)\leftrightarrow *. $$ This correspondence holds for any $\operatorname{Spec} k[x]$ with algebraic closed field $k$.   When $k$ is not algebraic closed, how should one understand $\operatorname{Spec} k[x]$? In the LaTeX document pasted below, I still see page numbers upon compiling, even though I've included a \pagestyle{empty} command below the \maketitle command. Any ideas what I'm doing wrong? Thanks in advance.  \title{Thesis Abstracts} \author{         Kurt Han Peek } \date{\today}  \documentclass[a4paper,11pt]{article} \usepackage{graphicx} \usepackage[margin=2cm]{geometry}  \begin{document} \maketitle \pagestyle{empty}  \section*{An Analysis of the Effects of Digital Phase Errors on the Performance of a FMCW-Doppler Radar} In modern frequency-modulated continuous-wave (FMCW) radars, the transmitter is increasingly being implemented using direct digital chirp synthesis (DDCS), which provides improvements in sweep linearity, stability, precision, agility, and versatility over analog techniques. Its main limitations are errors due to sampling of the modulating signal, phase truncation, and digital-to-analog converter (DAC) quantization, which produce spurious signals due to their deterministic nature. This thesis presents an analysis and simulation of the effect of two sources of digital phase errors -- namely, sampling of the modulating signal and phase truncation -- on the performance of a FMCW-Doppler transceiver employing a DDCS in its transmitter. An upper bound for the amplitude of spurious targets resulting from these digital phase errors is established. Further, it is shown that provided the phase errors are periodic with the sweep period, the spurious targets are not offset from the target in Doppler. An algorithm for selecting the digital chirp parameters of a DDCS so as to ensure periodic and phase-continuous sweep transitions, which guarantee periodic phase errors and reduce `spectral splatter', is devised. Finally, we investigate parallels of FMCW radar in the optical domain, and consider the fundamental question whether range resolution is fundamentally limited by the bandwidth of a transmitted signal, or its carrier frequency.  \section*{Estimation and Compensation of Frequency Sweep Nonlinearity in FMCW radar} One of the main issues limiting the range resolution of linear frequency-modulated continuous-wave (FMCW) radars is nonlinearity of frequency sweep, which results in degradation of contrast and range resolution, especially at long ranges. Two novel, slightly different, methods to correct for frequency sweep nonlinearity by digital post-processing of the deramped signal were introduced independently by Burgos-Garcia et al. (Burgos-Garcia, Castillo et al. 2003) and Meta et al. (Meta, Hoogeboom et al. 2006). In these publications, however, no formal proof of the techniques was given, and no limitations were described. In this thesis, we prove that the algorithm of Meta is exact for temporally infinite chirps, and remains valid for finite chirps with time-bandwidth products which are large compared to unity, provided the maximum frequency component of the phase error function is small compared to the square root of the nominal chirp rate. It is also shown that the algorithm of Meta reduces to that of Burgos-Garcia in this limit. A digital implementation of both methods described. Finally, we propose a novel method to measure the systematic phase errors which are required as input to the compensation algorithm.  \end{document} Properties of the greatest common divisor: $\gcd(a, b) = \gcd(a, b-a)$ and $\gcd(a, b) = \gcd(a, b \text{ mod } a)$ Prove that (a) gcd(a, b) = gcd (a, b – a) (b) Let r be the remainder if we divide b by a. Then gcd(a, b) = gcd(a, r).  I solved part a like: Assume a=pcommonpa b=pcommonpb gcd (a,b) = pcommon b-a = pcommon*(pb-pa)  So: gcd(a, b-a) = common again But I need help for part b guys I can solve it but I need a mathematical expression. to infinitive as direct object?! (as in : I promised to get there in time.) I'm a passionate Korean English-learner, and I would like to ask you a question about grammar.   It was 3 years ago when I first started to study the to-infinitive as studying grammar with Grammar In Use series and English Grammar series by Longman, having been exposed to English for 3 years. There wasn't any mention of the author saying that the usage of the to-infinitive is as an (direct) object. And for me also, the times I have encountered sentences in which the usage of the to-infinitive, so to speak, as direct objects were used, I've read them as some kind of obligatory adverbial that shows what the verbs are about, not as objects, which means they are noun phases! 100% of Korean textbooks say the usage is as a direct object, but I doubt it. I think it's some kind of obligatory adverbial that shows what the verbs are about.    I thought it was better for me to ask Americans themselves. Do you read this usage of to-infinitive as a noun phrase or an adverbial? Is there a practical way to change the phone number in Uber app when arriving at a country? I just read  on EE.SE and that triggered a doubt in me.  If I were asked whether there is any relationship between the electrical resistivity and the density of a substance I'd answer negatively, but is it really so?   Is there really no (relevant) relationship between those two bulk properties of a material? Can anyone confirm or disprove this and give some explanation thereof? MySQL cartesian product between two SELECT statements What is the meaning of this senctence:Excellent answer. Delusions of grandeur. Folks have been institutionalised for less Can an atheist at least agree that existence is a mystery that is bigger than all of us?                                                       You’ve got it backwards. It’s religious people who try to fill the mysteries of the universe with beliefs based on nothing emprical. Where atheists say “I don’t know” religious people say “It’s God.” Atheists are comfortable with not knowing everything. They know humans are infinitesimal specks in the universe. Religious people think the entire universe was created for and about them. Talk about grandiosity. Comment:Delusions of grandeur. Folks have been institutionalised for less lol Blank questions and answers I have white blank on the posts.    App Version: 1.0.89 Device Manufacturer: Meizu Device Model: m2   OS Version: 5.1 (1464689330)    I can't see the questions and the answers and the comments.  I saw that it's a common issue..  Does it have a fix?   EDIT -  I saw  answer. This answer is ~2 years old. It says that it will fix in the next update.  His version is 1.0.59 - mine is 1.0.89. Huge different.   The problem is still exists! Why can't the indefinite integral $\int\frac{\sin(x)}{x}\mathrm dx$ be found? The Software Center in Ubuntu 16.04 (Gnome Software) shows only part of the packages available. While e.g. apt list qtcreator correctly lists the corresponding package a search in Software Center finds nothing.  Is there something misconfigured? I upgraded from 15.10. Until now I have used the old Ubuntu Software Center which worked fine. Does copying a spell from a scroll into your spellbook consume the scroll? 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