What is the easiest way to find the gcd of two polynomials in the ring $\mathbb{Z}/m\mathbb{Z}$? ($m$ is prime)
Why does the Euclidean algorithm work for polynomials? What is the proof?
Sum data based on cell color in Google Spreadsheets I have seen  and altough I face the same problem, the answer to that question is not helpful to my case.   I have large spreadsheets which are changed frequently. So, according to the solution accepted to that question, it is hard to duplicate all populated columns and it is cumbersome to edit one more cell everytime I wish to make a change on cells background colors.  Isn't there a way to sum data based on a condition regarding cell background color in Google Spreadsheets? I'm trying to convert some strings that are in French Canadian and basically, I'd like to be able to take out the French accent marks in the letters while keeping the letter. (E.g. convert é to e, so crème brûlée would become creme brulee)  What is the best method for achieving this? If a sequence converges in a metric space, it is Cauchy, and in $\mathbb{R}^k$ every Cauchy sequence converges.   Therefore, in $\mathbb{R}^k$ a sequence converges iff it is Cauchy.  Let $\{s_n\}$ be a sequence in $\mathbb{R}$ where each $s_n=\sum_{k=1}^na_k$. Therefore, by the above, every series converges iff  $$\left | \sum_{k=m}^n a_k\right| &lt;\epsilon$$ For a given $\epsilon &gt;0$ and an integer $N$ such that $N\le m\le n$. If $n=m$ then the statement reduces to:  A series converges if and only if $$|a_n| &lt; \epsilon $$ For a given $\epsilon &gt;0$ and an integer $N$ such that $N\le n$.  This clearly cannot be (e.g Harmonic series). When does the equivalence become an implication. Let $a=p_1^{x_1}p_2^{x_2}\cdots\cdot p_q^{x_q}$ and $b=p_1^{y_1}p_2^{y_2}\cdots p_r^{x_r}$ ,   say — Since gcd(a,b) is the largest common divisor of a and b and is divisible by any other common divisor of the two,   GCD$(a,b)=p_1^{\min(x_1,y_1)}p_2^{\min (x_2,y_2)} \cdots p_q^{\min (x_q,y_q)}$  LCM$(a,b)= p_1^{\max(x_1,y_1)}p_2^{\max(x_2,y_2)}\cdots p_r^{\max (x_r,y_r)}$  I understand GCD(a,b) has to divide both $a,b$. Therefore the exponent of any prime factor $p_i$ in GCD has to be in both $a,b$  — therefore $p_i^{\min(x_i, y_i)}$. But I'm muddled and anxious. Why does $\min$ appear in GREATEST common divisor? Why does $\max$ appear in LOWEST common multiple? Confusion on Derived Sets and the $n$-the Derived Set I want to solve the following exercise 1.3.C. from page 27 of R. Engelking: General Topology.  For every positive integer $n$ the $n$-th derived set $A^{(n)}$ of a subset $A$ of a topological space $X$ is defined inductively by the formulas: $$  A^{(1)} = A^d \quad \textrm{and} \quad A^{(n)} = (A^{(n-1)})^d. $$ (a) Give an example of a set of real numbers that has three consecutive derived sets distinct from each other.  (b) Give an example of a set of real numbers that has infinitely many derived sets distinct from each other.   The definition is $$   A^d := \{ x \in X \mid x \in \overline{A \setminus \{ x \}} \}. $$ I am confused, cause the set $A^d$ ist closed, i.e. $\overline{A^d} = A^d$. And closed means it already contains all its limit points, so how can than $(A^d)^d$ be different from $A^d$? Windows Subsystem for Linux (WSL): what can't I do with the Ubuntu application for Microsoft Windows? Harry: "He made seven Horcruxes?"   Dumbledore: "...Not seven Horcruxes: six. The seventh part of his soul, however maimed, resides inside his regenerated body..." (HP6, US Hardback, page 503).    I would think that Dumbledore had already known that there were 6 horcruxes:    the diary  Gaunt's ring  Salazar's locket Helga Hufflepuff's cup Nagini (the whole essence divided thing)   Now those are the five that he was sure of. He knew that when Riddle came to Hogwarts to ask for the DADA position, he was really there so he could hide a horcrux. (But he never got a chance to tell Harry that, right?) Did Dumbledore know of the diadem? Or did he just think Riddle wanted to hide the cup?   Did Dumbledore say there were 6 and not 7 because he didn't know about the diadem but did know about Harry being a horcrux? Or did he say there were 6 to ensure Harry didn't guess he was a horcrux? Sorry if this makes no sense.  I know that in the end there were 7 horcruxes and an 8th part of Voldemort's soul right? How to emulate titletoc with tocloft I was using titletoc until I realized that it has some incompatibilities with hyperref. The thing is, that titletoc can do the same thing in a easier way, less one.  \titlecontents*{subsection}[5.4em] {\small} {\thecontentslabel~ } {} {} [.~\textbullet\ ] [.]   this part does the subsection as a paragraph (with section number and without page number). I have realized that the problem is the pseudo-code used by tocloft for the entries in the TOC  {\cftXfont {\cftXpresnum SNUM\cftXaftersnum\hfil} \cftXaftersnumb TITLE}% {\cftXleader}{\cftXpagefont PAGE}\cftXafterpnum\par   I mean, the last command \par make impossible to leave two subsection entries in the same textline. Any idea? Another interesting solution is with etoolbox, some examples with that package are awesome, but it's not on my hands.  This is a MWE:  \documentclass{book}  \usepackage{titletoc} \titlecontents*{subsection}[5.4em]{\small} {\thecontentslabel~ }{}{}[.~\textbullet\ ][.]  \begin{document}  \tableofcontents  \chapter{Test Chapter 1} \section{Test Section 1.1} \subsection{First Test Subsection} \subsection{Second Test Subsection} \subsection{Third Test Subsection} \subsection{Fourth Test Subsection} \section{Test Section 1.2} \subsection{First Test Subsection} \subsection{Second Test Subsection}  \chapter{Test Chapter 2} \section{Test Section 2.1} \subsection{First Test Subsection} \subsection{Second Test Subsection} \subsection{Third Test Subsection} \subsection{Fourth Test Subsection} \section{Test Section 2.2} \subsection{First Test Subsection} \subsection{Second Test Subsection}  \end{document} How to create dynamic JSF form fields Do continuous linear functions between Banach spaces extend? Why did the limit of power change from wizard/witch to wand? So in the first 5 or 6 books, it was very clear that the wand chooses the wizard and the power/level of the spells the wizard produces is based on the knowledge and skill of the wizard.  This is referenced by people constantly being telling Harry that he wasn't skilled enough to perform certain spells, or Bellatrix telling Harry that he needed to mean the Cruciatus curse for it to be effective.  However, in the 7th book, this entire idea is changed to the wand being the thing that's powerful and regardless of the wizard's/witch's skill or knowledge, he/she could still perform a very advanced spell if it's something the wand could handle.  Goyle accidentally creating a fire serpent is something that instantly comes to mind from the movies(I don't remember if that happened in the book).  So, my question is, what actually holds the power, the wizard/witch, or the wand? I have an argument with the conclusion: 'The sky is blue and it is not the case that the sky is blue'.  Without knowing the premises of this argument, am I right in saying this argument is invalid since the this conclusion is necessarily false? What is the cause of gravitional force of attraction? Ok,this is a silly question. But for quite long I have been thinking about this. What is gravitation all about?? My book directly writes since force of gravitation is directly proportional to the masses and inversly proportional to the square of the distance ie. $$F = G\frac{M_1 . M_2}{r^2}$$ . So is this gravitation all about? What is the cause of it? The book then says     We are not in the scope in classical approach so as to discuss the cause of this. However,as a matter of fact that if there are more than one mass,then there is gravitation as it is their fundamental property.( Cause of gravitation can be attributed to exchange of non-classical particles between the masses) .    Then it ended the talkings. What are those non-classical particles? Why and how are they exchanged? So far I have studied physics , I never found such weird phenomenon like this. I want to know what is the cause of gravitation and what is the cause of exchange of those so-called non-classical particles? And how do they do it? 408 Request Time-outs still continues I'm frequently getting 408 Request Time-out errors  when browsing through Stack Overflow and Meta. I used to get this four or five times a day before, but now it has become very frequent. I've read  in which Kyle Brandt ♦ says:     We send our web logs to SQL so we can query them. It turns out that we have a bug that prevents 408s from being logged to SQL. [...] I will do more extensive research when our 408s are part of queryable logs (should be a week or two [...]).   My internet connection is somewhat stable and I haven't experienced this issue on other sites before. I don't think it's a browser-specific issue as it is reproducible on latest versions of Google Chrome and Firefox (on Ubuntu 13.10). However, it is specific to Stack Overflow and Meta. I'm unable to reproduce this on other SE sites.  Am I the only one facing this issue? Could it be something wrong with the regional CDN or something like that? Could someone please take a look at this issue? ()  This appears when I try to close-vote questions, too: Recommended books for advanced undergraduate electrodynamics How to prove polynomials with degree $n$ does not form a vector space? This is one of my linear algebra problems:     Prove that polynomials of degree $n$ does not (The professor made these words bold intentionally) form a vector space.   From what I read, the set of polynomials of degree $n$ should be a vector space, because:   There is an "One" and a "Zero" in this set; We can find inverse for addition and multiplication from this set; It follows all the axioms of addition. It follows all the axioms of scalar multiplication.   Then can someone give me some hints to prove it does not form a vector space? We know that in high temperature, resistivity in metals goes linearly with temperature. As temperature is lowered, resistivity goes first as $T^5$ due to "electron-phonon" interaction, and then goes as $T^2$ as temperature is further lowered due to "electron-electron" interactions.   My question is, is there an intuitive way of understanding the above power-law dependence at each temperature range physically? Are there any differences between insmod and modprobe? Codomain of function complete implies function space complete Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces.   Let $(C_b(X,Y), d_C)$ be a function space with the metric $$d_C(f,g):=\sup\limits_{x\in X} d_Y(f(x), g(x)).$$  Prove that if $(Y,d_Y)$ is complete then $(C_b(X,Y), d_C)$ is complete.  Unfortunately, I have no idea how to approach this problem. I can think of something like this:  Let $(y_k)_k\subset Y$ be a Cauchy sequence converging to $y\in Y$. Then there exists some $f\in C_b(X,Y)$ such that $f(x_k)=y_k$ for all $k\in \mathbb N$. Since $f$ is continuous, $\lim\limits_{k\to\infty} d_Y(f(x_k), y)=0$.  But how do I go from here? As far as I understand, what needs to be proved is that any Cauchy sequence of functions in $C_b(X,Y)$ converges in $C_b(X,Y)$, that is, that the limit is bounded and continuous. But how do we devise such a sequence of functions? TZInfo::DataSourceNotFound error starting Rails v4.1.0 server on Windows