Merging seemingly contiguous polygons leaves artifacts
Removing small spaces (slivers) between polygons?
Is there a tool to update my PPA sources.list entries after upgrading? How do I read image files directly from a zip without extracting to disk? Freestyle Background color not white? I'm following  and I've set my compositor up in the exact same way as he did, except my background is like a slightly light grey and I don't know why.  In his video's he's getting a bright white as his background and I can't see where I'm going wrong.  Here's the render:   Here's the compositor window:    file.  Please help! I really want to know why it's not happening. Exports look fine but when printing on plotter, text disappears, using ArcMap I exported a map and everything looks fine when viewing the PDF. But when printing it on my plotter, all the text disappears. The scale bar would bring but the text associated to the scale bar wouldn't. The title block also disappears. I've tried messing around with the settings. I've checked/unchecked &quot;Embed All Document Fonts&quot; and &quot;Convert Marker Symbols to Polygons&quot;. I'm not sure what I'm doing wrong here. What's weird is I exported this map a few weeks ago and it printed just fine. Today, I needed to make a change to the map and exported it out, and now I am experiencing this issue. I must have changed one of the export settings in the past few weeks but I'm not sure what it is. Has anyone experienced this? Why isn't the gamma function defined so that $\Gamma(n) = n! $? As a physics student, I have occasionally run across the gamma function  $$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$  when we want to generalize the concept of a factorial.  Why not define the gamma function so that   $$\Gamma(n) = n!$$  instead?  I realize either definition is equally good, but if someone were going to ask me to choose one, I would choose the second option.  Are there some areas of mathematics where the accepted definition looks more natural?  Are there some formulas that work out more cleanly with the accepted definition? I am trying to make a top down shooter that could potentially have hundreds of objects at once at one point. I need to check collision with projectiles, walls, and enemies. I am currently using ArrayLists which probably won't suffice later on and was wondering if someone had a good method or knows a good method for checking all of those objects at once. This is going to be a game for smartphones and is built in java. Any ideas would be great or if you could point me in some direction that would be fantastic.  Thanks in advance! Let $A =$ {$(x,y,z) \in \mathbb{R}^3 |$  $x,y,z \geq 0$ and $x+y+z \leq 1$}.   Compute the integral $$\int_{A} x^m y^n z^p (1-x-y-z)^q dx dy dz$$  using new variables $u = x+y+z$, $v=\frac{y+z}{x+y+z}$, $w=\frac{z}{y+z}$.  Also, although it was not stated in the problem, I'm assuming that $m,n,p,q \in \mathbb{Z}$.   Since $f(x,y,z)=(x+y+z, \frac{y+z}{x+y+z}, \frac{z}{y+z}) = (u,v,w)$, $f^{-1}(u,v,w) = (u(1-v), uv(1-w), uvw)$  Thus J $= \begin{vmatrix} 1-v &amp; -u &amp; 0\\  v(1-w) &amp; u(1-w) &amp; -uv\\  vw &amp; uw &amp; uv\\ \end{vmatrix} = u^2v$  Skipping the part where I find the new limits of integration, and skipping some algebraic simplification we have:  $$\int_{A} x^m y^n z^p (1-x-y-z)^q dx dy dz$$   $$= \int_{0}^1 \int_{0}^1 \int_{0}^1 u^{m+n+p+2}(1-u)^q v^{n+p+1}(1-v)^mw^p(1-w)^n du dv dw$$  However, I have no  idea where to go from here. Any help would be much appreciated. Generalizing a sequence. Consider a sequence of the form:  $x[n]=-1,-1...,1,1,...,-1,-1...,1,1... \quad n&gt;0$  One can think of this as a square wave ($\pm1$) with a 50 % duty cycle (coming from EE).  For the simplest case of such a series i.e. $-1,+1,-1,+1,-1...$, a general equation is quite trivial ($x[n]=-1^n$).  Can we have e general equation for the other cases i.e. when there are multiple $-1$'s followed by $+1$'s ? Convert string to Title Case with JavaScript Is there a simple way to convert a string to Title Case? E.g. john smith becomes John Smith. I'm not looking for something complicated like , just (hopefully) some kind of one- or two-liner. This question is a result of having too much free time years ago during military service. One of the many pastimes was playing tic-tac-toe in varying grid sizes and dimensions, and it lead me to a conjecture. Now, after several years of mathematical training at a university, I am still unable to settle the conjecture, so I present it to you.  The classical tic-tac-toe game is played on a $3\times3$ grid and two players take turns to put their mark somewhere in the grid. The first one to get three collinear marks wins. Collinear includes horizontal, vertical and diagonal lines. Experience shows that the game always ends in a draw if both players play wisely.  Let us write the grid size $3\times3$ as $3^2$. We can change the edge length by playing on any $a^2$ grid (where each player tries to get $a$ marks in a row on the $a\times a$ grid). We can also change dimension by playing on any $a^d$ grid, for example $3^3=3\times3\times3$. I want to understand something about this game for general $a$ and $d$. Let me repeat: The goal is to make $a$ collinear marks.  I assume both players play in an optimal way. It is quite easy to see that the first player wins on a $2^d$ grid for any $d\geq2$ but the game is a tie on $2^1$. The game is a tie also on $3^1$ and $3^2$, but my experience suggests that the first player wins on $3^3$ but the game ties on $4^d$ for $d\leq3$. It seems quite credible that if there is a winning strategy on $a^d$, there is one also on $a^{d'}$ for any $d'\geq d$, since more dimensions to move in gives more room for winning rows.  to a related question tells that for any $a$ there is $d$ so that there is a winning strategy on $a^d$.  This brings me to the conjecture:     There is a winning strategy for tic-tac-toe on an $a^d$ grid if and only if $d\geq a$. (Refuted by TonyK's answer below.)   Is there a characterization of the cases where a winning strategy exists? It turns out not to be as simple as I thought.  To fix notation, let $$ \delta(a)=\min\{d;\text{first player wins on }a^d\} $$ and $$ \alpha(d)=\max\{a;\text{first player wins on }a^d\}. $$ The main question is:     Is there an explicit expression for either of these functions?   Or decent bounds?   Partial answers are also welcome.   Note that the second player never wins, as was discussed in .    A remark for the algebraically-minded: We can also allow the lines of marks to continue at the opposite face when they exit the grid; this amounts to giving the grid a torus-like structure. Now there are no special points, unlike in the usual case with boundaries. Collinear points on a toric grid of size $a^d$ corresponds to a line (maximal collinear set) in the module $(\mathbb Z/a\mathbb Z)^d$. (If $a$ is odd, then $a$ collinear points in the mentioned module add up to zero, but the converse does not always hold: the nine points in $(\mathbb Z/9\mathbb Z)^3$ with multiples of three as all coordinates add up to zero but are not collinear.) This approach might be more useful when $a$ is a prime and the module becomes a vector space. Anyway, if this version of the game seems more manageable, I'm happy with answers about it as well (although the conjecture as stated is not true in this setting; the first player wins on $3^2$). Given two points (defined, for instance using nodes), I want to compute the distance between them.   Is there some build in functionality in tikz to do this? If not, how can it be done using the mathematical engine?   The application I have in mind is to draw an circular arc centered at (a) and passing through some second point (b), where only (a) and (b) are known. Is it correct to use that/which in the relative clause when it is referring to the place as an object pronoun? I was teaching an ESL class and came across this sentence which confused me a lot:  &quot;The Queen Alice is the place that I go to the most&quot; &quot;Isn't that the place which kids under 12 can't enter?&quot;  Are these 2 sentences correct? Shouldn't &quot;where&quot; be used for both sentences when it's referring to the place and replacing the object? Nested List Indices Distance between surfaces I'm on ub20LTS and I really dislike how much top space the topmost bars use. Any idea on how to remove or reduce their numbers?  Edit: Thanks to everybody's support, this is how my workspace looks like now. I've :  Installed gnome-extensions for firefox. Turned off menu bar in ff. Installed dash to dock. Installed hide top bar.  This way you can better utilize your screen space. How to show setCustomValidity message/tooltip without submit event When should I use \input vs. \include? There are two different commands to incorporate another file into the source of some document, \input and \include. When should I use one or the other? What are the differences between them? Are there more things like them to be aware of? How to set a shortcut for "Always on Top" for window? In 12.04 (Unity), I would like to assign a shortcut to the "always on top option" for a window. Probably something like Ctrl+Shift+Home to turn on and Ctrl+Shift+End to turn off, or whatever. Is there an option to do this somewhere in the Compiz Settings Manager?  This (old)  claimed that      For compiz, Enable Extra WM Actions. Check the properties, and insert ALT+A to Key field for Toggle Always-On-Top action.   But I couldn't find that. If $p,q,r$ are all primes,and $p|qr-1$,$q|pr-1$ and $r|pq-1$,find all possible values of $pqr$. If $p,q,r$ are all primes,and $p|qr-1$$~~~~~~~~~~~~~$$q|pr-1$$~~~~~~~~~~$ and $~r|pq-1$. Find all possible values of $pqr$.    My work: $qr-1=pk_1$ $pr-1=qk_2$ $pq-1=rk_3$ From the above equations, we can conclude that either $k_1,k_2,k_3$ are all even or one of the primes is $2$. I can also get that, $p^2q^2r^2=(pk_1+1)(qk_2+1)(rk_3+1)$ Now, I cannot proceed. Please help! No Surname in indian passport and given name is considered as last name in E ticket for an international flight I have booked tickets for my parents to visit Mauritius for a week and my mom's passport (given name is xxxx) doesn't have surname, so in the e-ticket form her name has been printed as "Mr xxxx". I have spoken to the travel agency and Air Mauritius, but didn't get any information regarding the issue.  Can anyone please tell me if there might be a problem boarding the flight?  I am very anxious right now since its their first international flight.