Hi Guys, I created the above bottle using the mirror modifier but when I am smoothing it, it leaves the edges/vertices joint, as I shown using red arrows. Please suggest how can I remove this visible separation line?
I was practicing with modeling a leaf using a image reference. Everything went fine and I finished modeling my leaf, but I wanted to make it more realistic (I'll add textures later), I saw lots of people using the Subdivision Surface modifier but it doesn't work for me.  My model:   My model when I add the Subdivision Surface modifier
So, if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$ means $1=ax+by$, and want to show $a^2x+b^2y=1$. By squaring $1=ax+by$ both sides, I get, $1=(ax)^2+b(2axby+by^2)$. It doesn't help my proof. Please help me with this proof. I have the following $n\times n$ matrix:  $$A=\begin{bmatrix} a &amp; b &amp; \ldots &amp; b\\ b &amp; a &amp; \ldots &amp; b\\ \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ b &amp; b &amp; \ldots &amp; a\end{bmatrix}$$  where $0 &lt; b &lt; a$.  I am interested in the expression for the determinant $\det[A]$ in terms of $a$, $b$ and $n$.  This seems like a trivial problem, as the matrix $A$ has such a nice structure, but my linear algebra skills are pretty rusty and I can't figure it out.  Any help would be appreciated. What are the requirements for a (beta) site logo? Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is , when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d&gt;1$?      — For instance if $p$ is prime (which is not coprime with $0$) then $$\mathbb{Z}[i]/(p) \cong \mathbb{F}_p[X]/(X^2+1) \cong \begin{cases} \mathbb{F}_{p^2} &amp;\text{if } p \equiv 3 \pmod 4\\ \mathbb{F}_{p} \times \mathbb{F}_{p} &amp;\text{if } p \equiv 1 \pmod 4 \end{cases}$$ (because $-1$ is a square mod $p$ iff $(-1)^{(p-1)/2}=1$).  — More generally, if $n=p_1^{r_1} \cdots p_m^{r_m} \in \Bbb N$, then  each pair of integers $p_j^{r_j}$ are coprime, so that by CRT we get $$\mathbb{Z}[i]/(n) \cong \mathbb{Z}[i]/(p_1^{r_1}) \times \cdots \times \mathbb{Z}[i]/(p_m^{r_m})$$  I was not sure how to find the structure of $\mathbb{Z}[i]/(p^{r}) \cong (\Bbb Z/p^r \Bbb Z)[X] \,/\, (X^2+1)$ when $p$ is prime and $r&gt;1$.  — Even more generally, in order to determine the structure of $\mathbb{Z}[i]/(a+bi)$ with $a+bi=d(x+iy)$ and $(x,y)=1$, we could try to use the CRT, provided that $d$ is coprime with $x+iy$ in $\Bbb Z[i]$. But this is not always true: for $d=13$ and $x+iy=2+3i$, we can't find Gauss integers $u$ and $v$ such that $du + (x+iy)v=1$, because this would mean that $(2+3i)[(2-3i)u+v]=1$, i.e. $2+3i$ is a unit in $\Bbb Z[i]$ which is not because its norm is $13 \neq ±1$.  — I was not able to go further. I recall that my general question is to known what $\mathbb{Z}[i]/(a+bi)$ is isomorphic to, when $a$ and $b$ are integers which are not coprime (for instance $a=p^r,b=0$ or $d=(a,b) = a^2+b^2&gt;1$).  Thank you for your help! Remove empty directory from CVS? I am trying to use notepad++ and regular expressions to edit a document. I have text in this format  123.45607894.165@abcd;aba 23.456@av1;084 9871.001@fab:9782581afa   And I need to get only the first part of the line , the one before the @ sign. How would I do that to get this:  123.45607894.165 23.456 9871.001 Inverse elements in the absence of identities/associativity. How does the comma operator work in C++?  For instance, if I do:  a = b, c;     Does a end up equaling b or c?   (Yes, I know this is easy to test - just documenting on here for someone to find the answer quickly.)  Update:  This question has exposed a nuance when using the comma operator.  Just to document this:  a = b, c;    // a is set to the value of b!  a = (b, c);  // a is set to the value of c!   This question was actually inspired by a typo in code.  What was intended to be  a = b; c = d;   Turned into  a = b,    //  &lt;-  Note comma typo! c = d; Hitting arrow keys adds characters in vi editor When I tried to use arrow keys in insert mode in vi editor the following characters are being inserted in the editor:   for ↓ I get B, for ↑ I get A, for ← I get D, for → I get C.   Please help me in resolve this problem. enumerate in multicols The following image illustrates what am looking for:    Essentially I want the numbering to occur as shown above but using the multicol and enumitem package. But when I try it by using the following code:  \documentclass{article} \usepackage{amsmath,amssymb} \usepackage{enumitem,multicol} \begin{document} \begin{enumerate} \item Evaluate each of the following if $x$ is a non-zero real number.     \begin{multicols}{2}     \begin{enumerate}[itemsep=1cm]% The item sep is my choice here.     \item $\dfrac{|x|}{x}$     \item $\dfrac{x}{|x|}$     \item $\dfrac{|-x|}{-x}$     \item $|x|-|-x|$     \end{enumerate}      \end{multicols} \end{enumerate} \end{document}   I get:    Essentially, I want the (b) to be where the (c) is. I know this has been done but I can't seem to find it. Custom Tools not Saving in Toolbar on Startup/ Normal.mxt? Complete Installation Guide for Android SDK / ADT Bundle on Ubuntu What all needs to be done to install Android SDK / ADT Bundle on Ubuntu?  Each of the steps in the answer was found on some different forum as I encountered obstacles getting started with my Android SDK and posted questions in appropriate forums thereafter. So I thought this would be useful in that it is a one stop solution available, in case one ever needs to re-install Ubuntu and install the Android SDK / ADT Bundle from scratch! While reading a question here whose answer was Stephen Baxter's &quot;Light of Other Days&quot; book, it reminded me of a short story I read with a similar idea. Probably read at least 30-40 years ago, in English. The fact that the Government has invented a device that can view the past is made public (by a whistleblower, I believe). Historians go nuts besieging the government to look at their favorite time periods to reveal the truth about various historical &quot;facts&quot; and or &quot;mysteries&quot; (Who shot JFK? Who was Jack the Ripper? Was Helen of Troy really that beautiful? Did Jesus exist? etc, etc.) And the government is stonewalls most of them and claims it's expensive to operate, hard to control to get exact time/location, etc. So someone (maybe the whistleblower?), knowing it's possible, does math/construction and builds his own version, and realizes it's easy to make, cheap to operate, and easy to use. So he makes plans to stick it to the government by releasing the info. Maybe he tries to force the government to release the info, or they realize he has made it, so they grab him and try to convince him to stop. But he is not convinced until they tell him, it can't see back very far (maybe just a few days/weeks, because it needs exponentially more power the further you look back.) Finally the agents tell him: &quot;Don't you see, you can use it to see what's happening just a few minutes ago, or even now. Your company can't make widgets? Zoom in on your competitors and see how they make them. Think your wife is fooling around on you? Zoom in on her and track her to see if she is. Your pretty neighbor is taking a bath now? Zoom in and watch her... These devices mean no one will have any privacy now.&quot; At which point he tells them it's already too late. They kept him there so long that the letters he mailed will already be delivered to various newspapers, scientists, historians, etc. Radio antenna producing waves in the visible spectrum If a radio could produce waves in the visible light spectrum, what would the result be?  This is a thought experiment that I've pondered for a few years now.  I realize there are a few/many real-world constraints, but if we lifted these constraints for the sake of thought, what could we expect?  Personally, I don't see why we wouldn't observe visible light emitting from the antenna, disregarding any light from Blackbody Radiation. For example,   If this van's a-rockin', don't come a-knockin'  (song title)   (song title)   I have a few related questions:   What is the "a-" or "-a" called? Is there a function beyond a rhythmic one? Why is "a-" attached to a word at all -- and why attach to a word at all? MySQL Conditional Insert How Do I get into the castle walls? Knitr: opts_chunk$set() not working in Rscript command When upgrading to Ubuntu 14.04 from Ubuntu 12.04 I am getting the following error:     W:Failed to fetch    Unable to find expected entry 'non-free/source/Sources' in Release   file (Wrong sources.list entry or malformed file) , E:Some index files   failed to download. They have been ignored, or old ones used instead. What fallacy dismisses problems by presenting "bigger" problems? Wasn't really sure how to phrase this, but I'm thinking of an instance in which someone diminishes a problem by presenting one of larger scope - as a rather shoddy example, "x political problem in America doesn't matter because half the world's population is starving."  I honestly have no idea if this is considered fallacious or not, but for some odd reason I can't shake the feeling that someone has described it to me as a fallacy before. Am I crazy or is there something to this (or something similar)?