This is a homework problem I have to solve, and I think I might be misunderstanding it. I'm translating it from Polish word for word.     $n$ points are placed on a circle, and all the chords whose endpoints they are are drawn. We assume that no three chords intersect at one point.      a) How many parts do the chords dissect the disk?      b) How many triangles are formed whose sides are the chords or their fragments?   I think the answer to a) is $2^n$. But I couldn't find a way to approach b), so I calculated the values for small $n$ and asked OEIS about them. I got . And it appears that there is no known formula for all $n$.  says that      $\text{triangles}(n) = P(n) - T(n)$      where $P(n)$ is the number of triangles for a convex n-gon in general   position. This means there are no three diagonal through one point   (except on the boundary). (There are no degenarate corners.)   This number is easy to calculate as:      $$P(n) = {n\choose 3} + 4{n\choose 4} + 5{n\choose5} + {n\choose6}        = {n(n-1)(n-2)(n^3 + 18 n^2 - 43 n + 60)\over720}$$      The four terms count the triangles in the following manner [CoE76]:      $n\choose3$: Number of trianges with 3 corners at the border.      $4{n\choose4}$: Number of trianges with 2 corners at the border.      $5{n\choose5}$: Number of trianges with 1 corners at the border.      $n\choose6$: Number of trianges with 0 corners at the border.      $T(n)$ is the number of triple-crossings (this is the number of   triples of diagonals which are concurrent) of the regular $n$-gon.   It turns out that such concurrences cannot occur for n odd, and,   except for obvious cases, can only occur for $n$ divisible by $6$.   Among other interesting results, Bol [Bol36] finds values of n   for which $4$, $5$, $6$, and $7$ diagonals are concurrent and shows that   these are the only possibilities (with the center for exception).      The function $T(n)$ for $n$ not divisible by $6$ is:      $$T(n) = {1\over8}n(n-2)(n-7)[2|n] + {3\over4}n[4|n].$$      where $[k|n]$ is defined as $1$ if $k$ is a divisor of $n$ and otherwise $0$.      The intersection points need not lie an any of lines of symmetry of   the $2m$-gon, e. g. for $n=16$ the triple intersection of $(0,7),(1,12),(2,14)$.   If I understand the text correctly, it doesn't give a general formula for $T(n)$. Also I've found a statement somewhere else that some mathematician wasn't able to give a general formula solving this problem. I haven't found a statement that it is still an open problem, but it looks like it to me.   So am I just misunderstanding the problem, or misunderstanding what I've found on the web, or maybe it is indeed a very hard problem? It's the beginning of the semester, our first homework, and it really scares me.
How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines have a common point. Further, this is not a non-Euclidean problem, but I wouldn't mind a discussion on the non-Euclidean nature of the problem.  I was thinking it might be easier to show using the unit sphere.    A GOOD REFERENCE I FOUND THAT WILL BE USEFUL TO ANYONE COMING ACROSS THIS PROBLEM IS Concrete Mathematics, Graham, Knuth, Patashnik pp. 4–8.
The summer training or (just) summer training Should we use the summer training or just summer training in the paragraph. As per me training is a noun and before singular noun we have to use an article and I am using the because I am talking about a specific training for engineers, I also know that before very common nouns we avoid articles, like bed, school, bus, university, so if you say we should use summer training then please give me a valid reason.     Company(name) has been providing the summer training for engineers   since 2010. We have been giving completely professional atmosphere for   engineers to complete the summer training with practical knowledge.   Our each faculty has a great experience in the education field. We   make sure you get all the important resources to make your summer   training a learning and an enjoyable summer training. You may call or   email us to get your confirmation about the training.   Thanks in advance. When should I use can? When should I use could? What is right under what context? All my college years I have been using public, and would like to know the difference between public, private, and protected?  Also what does static do as opposed to having nothing? How to know if something is a primitive concept, a law, a definition or a theorem Some basic Physics books are often misguiding in the sense that they don't make clear whether something is a primitive concept, a law, a definition or a theorem. This is often a little confusing. I've asked here some moments ago if there's a definition of force, and I've been redirected to a question about whether Newton's Laws are laws or definitions of mass and force.  This is just an example, but there are many others, like for instance, the conservation of energy, which some books present as a theorem, and some places says that this is a law of Physics. Time on the other hand, is something that according to Feynman's lectures on Physics, cannot be defined, so we just say how to measure it and leave it as a primitive concept.  In this context: how in general can we identify whether something is a primitive concept, a fundamental law of Physics, a definition or a theorem? Why is any pgr_* routing function taking forever based on OSM data in an pgrouting enabled DB Is the total energy of the universe zero? In popular science books and articles, I keep running into the claim that the total energy of the Universe is zero, "because the positive energy of matter is cancelled out by the negative energy of the gravitational field".  But I can't find anything concrete to substantiate this claim. As a first check, I did a calculation to compute the gravitational potential energy of a sphere of uniform density of radius $R$ using Newton's Laws and threw in $E=mc^2$ for energy of the sphere, and it was by no means obvious that the answer is zero!  So, my questions:   What is the basis for the claim &ndash; does one require General Relativity, or can one get it from Newtonian gravity? What conditions do you require in the model, in order for this to work? Could someone please refer me to a good paper about this? Whats the simplest way to edit and add files to "/var/www"? Having installed the web server is there a simple way to set a user able to use the graphic interface to copy files and directories to the local web server /var/www  I gave myself administrative privileges in Ubuntu but it still doesn't allow copies. Difficulty in creating macro '\newcommand' for the beginning and end of frames in Beamer Let $S=\{1,2,3,4,\dotsc,N\}$ and $X=\{ f: S \rightarrow S \mid x < y \Rightarrow f(x)\leqslant f(y)\}$. Then $|X|$ is equal to what? What is the best way to store fresh herbs like parsley or cilantro?  I find I have about the same luck with them whether they are left in a plastic bag or I put them in a cup of water. What are the pros and cons of an articulated display? How to understand the memory usage and load average in linux server I am using a linux server which has 128GB of memory and 24 cores. I use top to see how much it is used. Its output is pasted at the end of the post. Here are two questions:  (1) I see that each of the running processes occupies a very small percentage of memory (%MEM no more than 0.2%, and most just 0.0%), but how the total memory is almost used as in the fourth line of output ("Mem:  130766620k total, 130161072k used,   605548k free,   919300k buffers")? The sum of used percentage of memory over all processes seems unlikely to achieve almost 100%, doesn't it?  (2) how to understand the load average on the first line ("load average: 14.04, 14.02, 14.00")?   Thanks and regards!  Edit:  Thanks!  I also really like to hear some rough numbers based on used percentage of memory to determine if a server is heavily loaded, since I once became the one who cramed the server without understanding the current load.  Is swap regarded as almost the same as memory? For example, when memory and swap are almost of same size, if the memory is almost running out but the swap is still largely free, may I just view it as if the used percentage of memory + swap is still not high and run other new processes?  How would you consider together CPU or memory (or memory + swap) usage? Do you become worried if either of them reaches too high or both?  Output of top：  $ top    top - 12:45:33 up 19 days, 23:11, 18 users,  load average: 14.04, 14.02, 14.00 Tasks: 484 total,  12 running, 472 sleeping,   0 stopped,   0 zombie Cpu(s): 36.7%us, 19.7%sy,  0.0%ni, 43.6%id,  0.0%wa,  0.0%hi,  0.0%si,  0.0%st Mem:  130766620k total, 130161072k used,   605548k free,   919300k buffers Swap: 63111312k total,   500556k used, 62610756k free, 124437752k cached    PID USER      PR  NI  VIRT  RES  SHR S %CPU %MEM    TIME+  COMMAND  6529 sanchez   18  -2 1075m 219m  13m S  100  0.2  13760:23 MATLAB 13210 timothy   18  -2 48336  37m 1216 R  100  0.0   3:56.75 absurdity 13888 timothy   18  -2 48336  37m 1204 R  100  0.0   2:04.89 absurdity 14542 timothy   18  -2 48336  37m 1196 R  100  0.0   1:08.34 absurdity 14544 timothy   18  -2  2888 2076  400 R  100  0.0   1:06.14 gatherData  6183 sanchez   18  -2 1133m 195m  13m S  100  0.2  13676:04 MATLAB  6795 sanchez   18  -2 1079m 210m  13m S  100  0.2  13734:26 MATLAB 10178 timothy   18  -2 48336  37m 1204 R  100  0.0  11:33.93 absurdity  12438 timothy   18  -2 48336  37m 1216 R  100  0.0   5:38.17 absurdity 13661 timothy   18  -2 48336  37m 1216 R  100  0.0   2:44.13 absurdity 14098 timothy   18  -2 48336  37m 1204 R  100  0.0   1:58.31 absurdity 14335 timothy   18  -2 48336  37m 1196 R  100  0.0   1:08.93 absurdity 14765 timothy   18  -2 48336  37m 1196 R   99  0.0   0:32.57 absurdity 13445 timothy   18  -2 48336  37m 1216 R   99  0.0   3:01.37 absurdity 28990 root      20   0     0    0    0 S    2  0.0  65:50.21 pdflush 12141 tim       18  -2 19380 1660 1024 R    1  0.0   0:04.04 top  1240 root      15  -5     0    0    0 S    0  0.0  16:07.11 kjournald  9019 root      20   0  296m 4460 2616 S    0  0.0  82:19.51 kdm_greet     1 root      20   0  4028  728  592 S    0  0.0   0:03.11 init     2 root      15  -5     0    0    0 S    0  0.0   0:00.00 kthreadd     3 root      RT  -5     0    0    0 S    0  0.0   0:01.01 migration/0     4 root      15  -5     0    0    0 S    0  0.0   0:08.13 ksoftirqd/0     5 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/0     6 root      RT  -5     0    0    0 S    0  0.0  17:27.31 migration/1     7 root      15  -5     0    0    0 S    0  0.0   0:01.21 ksoftirqd/1     8 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/1     9 root      RT  -5     0    0    0 S    0  0.0  10:02.56 migration/2    10 root      15  -5     0    0    0 S    0  0.0   0:00.34 ksoftirqd/2    11 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/2    12 root      RT  -5     0    0    0 S    0  0.0   4:29.53 migration/3    13 root      15  -5     0    0    0 S    0  0.0   0:00.34 ksoftirqd/3 What is a word for deliberate dismissal of some facts? What happens when a creature with "Whenever a creature dies" dies simultaneously with multiple other creatures? has the following ability:     Whenever a creature you control with a +1/+1 counter on it dies, return that card to the battlefield under your control at the beginning of the next end step.   I had a number of creatures on the battlefield, including Marchesa herself, with one or more +1/+1 counters on them when  was played, destroying each creature on the board.  When Marchesa and several other creatures would be killed simultaneously,   Does Marchesa's ability trigger for herself? Does Marchesa's ability trigger for the other creatures under my control with +1/+1 counters?   In attempting to resolve the issue, we tried to look things up, leading to , which cites Rule 603.6d as proving that Marchesa's ability should trigger both for herself and for each other creature with +1/+1 counters.  Unfortunately, none of us could actually understand Rule 603.6d, aside from this example:     Two creatures are on the battlefield along with an artifact that has the ability Whenever a creature dies, you gain 1 life. Someone plays a spell that destroys all artifacts, creatures, and enchantments. The artifacts ability triggers twice, even though the artifact goes to its owners graveyard at the same time as the creatures.   The example certainly seems analogous to me, but without understanding the actual rule, rather than just the example, we weren't too sure.  A complete answer has to not only answer the question, but back up that answer with citation and explanation of the official rules on the matter. How to do volumetric lighting for a forest scene? What is the difference between "is subject to" and "is subjected to"? I found the following sentence on the U.S. Customs and Border Protection website:     All persons, baggage, and other merchandise arriving in or leaving the   United States are subject to inspection and search by CBP officers.   I would like to know whether changing "subject" to "subjected" will change the meaning of the sentence. I created the shiftcommandn shortcut for Firefox in Syst Prefs to match other browsers' behavior.     It registers in the Firefox Menubar.     But pressing the shortcut opens a new non-private window, and it loads there the first item in my "Top Sites".   I imagine I could create another shortcut to override this behavior, but what is this behavior? I don't see anything highlighted in the Firefox Menubar when I hit the shortcut. Or might there be another solution (other than using a different shortcut to open new private windows)?   Problem exists on both Mojave and High Sierra. Default Firefox shortcut (shiftcommandp) still opens a new private window. This problem is bugging me for some time.     $f:(X,\mathcal{T}) \to (Y,\mathcal{T}')$ is a continuous bijection where $(X,\mathcal{T})$ is compact and $(Y,\mathcal{T}')$ is $T_2$ (i.e. Hausdorff and $T_1$). Then show that $f$ must be a homeomorphism.   So far I have failed to understand any relation between the compactness of domain and $T_2$-ness of the range. I tried in vain to prove that $f(U)$ is open in $(Y,\mathcal{T}')$ whenever $U$ is so in $(X,\mathcal{T})$. I know $f$ makes $X$ and $Y$ respectively $T_2$ and compact and hence makes both of them $T_4$ spaces. But that is not coming to any use. I would be glad if someone helps me with a solution. Thank you. Whenever I open a .sh file, it opens it in gedit instead of the terminal. I can't find any option similar to Right Click → Open With → Other Application... → Terminal.  How do I open this file in the terminal? Difference between @interface definition in .h and .m file