Optimum Dimensions of a Closed Cylindrical Can
Height/Radius ratio for maximum volume cylinder of given surface area
Correct usage of "lest" Which of the followings is/are the correct usage of the word "lest"? How are these different from each other?   ...hesitant to speak out lest he be fired. ...hesitant to speak out lest he'd be fired. ...hesitant to speak out lest he would/will be fired.   P.S. I took point number 1 as the reference for this question from the definition of lest from the Merriam-Webster Dictionary. Conjugacy Class in a matrix group with columns of standard basis I want to find the conjugacy classes of this group: $$G=\{(a_1,a_2,a_3):a_i=\pm e_j\}\subset GL_3(\mathbb C)$$ I know that $S_3$ is one of the subgroups of this group, so I can check which elements of $S_3$ continues to be conjugate in our group, I know that in $Z(G)$ there is two elements $I,-I$. I know that I have to check the diagonal matrices on which ones are conjugate, but I don't know how to finish and find the others conjugacy class of this group, and I don't Know how to find the number of the elements of the conjugacy class. I'll be thankful for help Difference Boolean modifier. How do I get it to work with my mesh? I want to make a screw head, but I can't get the cross in the screw head; the Boolean modifier doesn't work.  Link: Today, when looking up the definitions of the phrasal verb "take sth up", I noticed that one of them was written like this: "to start or begin something such as a job." For the first time I realised that probably "start" and "begin" don't exactly mean the same thing.So the question is what the difference is between the two, and if there are cases when they mean exactly the same thing. Though tangentially related to another post on MathOverflow (), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but...    Suppose $\Sigma$ is a sigma-algebra over a set, $X$. For any given topology, $\tau$, on $X$ denote by $\mathfrak{B}_X(\tau)$ the Borel algebra over $X$ generated by $\tau$.     Question 1. Does there exist a topology, $\tau$, on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$?   If the answer to the previous question is affirmative, it makes sense to ask for the following too:     Question 2. Denote by ${\frak{T}}_X(\Sigma)$ the family of all topologies $\tau$ on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$ and let $\tau_X(\Sigma) := \bigcap_{\tau \in {\frak{T}}_X(\Sigma)} \tau$. Is $\Sigma = \mathfrak{B}_X({\frak{T}}_X(\Sigma))$?   Updates. Q2 was answered in the negative by Mike (). Why is vote time missing in the SE data dump and SEDE? How can I tell how long (more or less) it will take me to complete a game? I'm going down my huge backlog of unplayed Steam games and I'm thinking the best way to tackle the list is to go through the shortest games first and then concentrate on the 20+ hour ones. I'm wondering if there's a website or something else that has an estimated amount of gameplay time for games.   Is there anything like that? Using HTML5/Canvas/JavaScript to take in-browser screenshots Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$ I'm learning calculus, specifically derivatives and applications of MVT, and need help with the following exercice:      Show that, for all $n &gt; 1$ $$\frac{1}{n + 1} &lt; \log(1 + \frac1n) &lt; \frac1n.$$    I tried to follow the below steps in order to prove the RHS inequality:   Proving that $f &lt; g$ on $I$ from $a$ to $b$:  Step $1$. Prove that $f' &lt; g'$ on $\operatorname{Int}(I)$.  Step $2$. Show that $f(a) \leq g(a)$ or that $f(a^+) \leq g(a^+)$  Following the above steps, let $f(x) = \log(1 + \frac1x)$ and $g(x) = \frac1x$, for all $x &gt; 1$. One has  $$f'(x) =  -\frac{1}{x^2 + x} \quad \text{and} \quad g'(x) = -\frac{1}{x^2}.$$  We note that, for every $x &gt; 1$, $f'(x) &gt; g'(x)$. Moreover, $f(1^+) = \log(2) &lt; 1 = g(1^+).$   My problem is that I got the wrong inequality sign in Step $1$.    Looking at the solution in my textbook, the author suggests using the MVT but I don't know how to apply it in this case. What will happen to my reputation when the Community Wiki status of a post is reversed? 2D graphics - why use spritesheets? I have seen many examples of how to render sprites from a spritesheet but I havent grasped why it is the most common way of dealing with sprites in 2d games.  I have started out with 2d sprite rendering in the few demo applications I've made by dealing with each animation frame for any given sprite type as its own texture - and this collection of textures is stored in a dictionary. This seems to work for me, and suits my workflow pretty well, as I tend to make my animations as gif/mng files and then extract the frames to individual pngs.  Is there a noticeable performance advantage to rendering from a single sheet rather than from individual textures? With modern hardware that is capable of drawing millions of polygons to the screen a hundred times a second, does it even matter for my 2d games which just deal with a few dozen 50x100px rectangles?  The implementation details of loading a texture into graphics memory and displaying it in XNA seems pretty abstracted. All I know is that textures are bound to the graphics device when they are loaded, then during the game loop, the textures get rendered in batches. So it's not clear to me whether my choice affects performance.  I suspect that there are some very good reasons most 2d game developers seem to be using them, I just don't understand why. TL:DR Smart guides aren't accurate - even though the snap, (end up causing more problems in the long run) is there a way to fix it?  Maybe there are some settings that I'm missing (need to be changed)    I use my smart guides all the time (and am overall pleased with them). I do have my issues with it, as explained here:  A lot of times, when I'm trying to make a bunch of triangles like this:    Done by making a triangle, cloning, flipping, rotating etc. Repeat until full  (Perfect) Snapping using Smart Guides    Ends up looking likebetween the triangles (Outline Mode)    Now I personally (with a strong attention to detail) will notice that there is something off after making just 1 or 2 triangles (Even before outline mode... strokes look oh so slightly thicker/off)   And then I check in outline mode... and sure enough:    Notice the line spaces (and this is in outline mode, so if I didn't notice it before, now it's sitting right in front of my nose)  The way I (seemingly) "fixed" the first image was by adding in a thick stroke outline so that masks out the difference, however, in a lot of my cases I don't want to have a stroke at all (and then you really notice the problems). PHP MySQLi doesn't recognize login info Is there a standard function in C that would return the length of an array? How do I deploy an OpenStack cloud with Juju? I'd like to get a test cloud up and running in preparation for a possible deployment of 12.04. I messed with the UEC a bit, but have had a much harder time finding anything helpful as a primer for the new stack. There is generic documentation on Diablo, but there are likely to be a lot of pieces already handy in Ubuntu that a specific guide could save time with.  Also, as much as I would love to, my budget doesn't include the $4-6k to get a small cluster running with the Canonical support. Simplify the expression $\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots +\binom{n+k}{k}$ Search among posts you've voted for Related posts:    These posts are different from mine in that they merely ask for the &quot;votes&quot; tab back or ask if it is at all possible to get such information, while I propose how to implement this as a search feature. Background I recently found myself facing a problem very similar to one that I had solved before. I remembered that I found the solution on SO and I remembered upvoting the post containing the solution. I wasn't able to find that particular post by just searching with relevant keywords. It would have been nice if I could search only among the questions I have voted on. I searched on meta and found that there had been a tab for looking at what you've voted on, but that it was removed due to lack of interest. This is understandable, since just listing the hundreds/thousands of posts you've voted on isn't particularly useful. Proposal What I want is to be able to search for posts I've voted on similarly to the way I'm able to search on my posts within a specific tag. For instance: user:310121 [iphone] apple  This will return all my posts tagged iPhone, where I mention Apple. This is brilliant. This is just how I want search to work. Now what I want to do is to extend this to include votes. The exact syntax isn't relevant, but I had to make one up to show you what I mean: To show all posts you've voted on: user:310121 {votes}  To show all posts you've upvoted: user:310121 {upvotes}  To show all posts you've downvoted: user:310121 {downvotes}  You could probably omit the the user part, since you're only allowed to see your own votes. (Unless it's desirable to be able to look at other peoples votes. Maybe you should be allowed to do this if you have enough rep? Well, that's another discussion.) I guess that even if you're allowed to see other peoples votes it would make sense to display your own votes if the user part is omitted. So in my case I could have done the following search and found the post I was looking for: {upvotes} keyword  Lastly I just want to suggest that clicking on your vote count will work just like clicking on a tag. If I click on a tag in my profile the following search will be done: user:310121 [tag]  So if I click on my upvote count the following search should be made: user:310121 {upvotes}  The same goes for downvotes: user:310121 {downvotes}  This would be a very useful feature and it wouldn't be intrusive in any way. The users that don't care for this feature might never know that it's there. And just for the record the favorite system doesn't replace this feature. How do I make animations a steady speed? When I make an animation, it always starts out slow, then speeds up, then slows down at the end.  I cannot figure out how to fix this. How to get the current date without the time? Problem after update: blue screen with only the mouse cursor