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What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters? Books for a beginner I am a student of 11th grade and i have completed the syallabus of both 11th and 12th grade maths with complete understanding and it was possible coz of the love for this subject that i have. I don't want to sound "larger than life" here, but my curiousity has now increased to learn maths at a deeper level. As of now i am reading "Linear algebra done right" by Axler. Thats an amazing piece of work and now i truly understand what "matrix" is all about. I also want to extend my knowledge on the following topics :  1) Algebra (groups,subgroups,homomorphisms etc)   2) Analysis   3) Geometry/Topology  Why i am here is because i wanted you guys to recommend a book on each of the above topic that would be appropriate for a beginner like me (I must mention here I have been working on Apostol's volume 1 calculus, thats a great text, but for a beginner, its best if the concept is explained in a broad manner and in as simple and easy words as possible) . I just want a book which explains the concept broadly rather than coming to the conclusion directly (which is not a great sight for a beginner like me) .   Having said that, no book is complete i understand. Thats why i am asking your recommendation as you guys are aware which text would be the best to start with for a beginner. I have searched on the net about this, but there are dozens of works available, and out of them all , i want the one which is the best (approximation) for a beginner .   Please don't misunderstand me in any way. I am just confused which text to go for, and i understand the books which u may suggest may be for undergraduate level course as these topics are for undergraduate level courses, thats not a problem at all . I need a easy to learn (i mean easy in terms of "broad" explanation) book. I hope you guys don't mind me questioning such a question on this forum . Thanks for all the help you guys have been providing me on this forum . Maths Maths Maths... the world is beautiful coz of u! :)) Triangle Inequality for norm integral $\|f\|_1=(\int_a^b [|f|^2+|f'|^2]\mathsf dx)^{1/2}$. Define $C_1^1[a,b]$ to be the space of continuously differentiable functions on $[a,b]$, with norm $$\|f\|_1=\left(\int_a^b[|f|^2+|f'|^2]\mathsf dx\right)^{1/2}.$$ Show that this is a proper definition of a norm. Is this normed space complete?  I'm stuck on the triangle inequality part. I'm having a hard time showing that $$\|f+g\|_1\leq \|f\|_1+\|g\|_1.$$ Any hints or solutions are greatly appreciated. Why AUC =1 even classifier has misclassified half of the samples? I am using a classifier which returns probabilities. To calculate AUC, I am using pROC R-package. The output probabilities from classifier are:  probs=c(0.9865780, 0.9996340, 0.9516880, 0.9337157, 0.9778576, 0.8140116, 0.8971550, 0.8967585, 0.6322902, 0.7497237)   probs shows probability of being in class '1'. As shown, the classifier has classified all of samples in class '1'.  True label vector is:  truel=c(1, 1, 1, 1, 1, 0, 0, 0, 0, 0)   As shown, classifier has misclassified 5 samples. But, AUC is:  pROC::auc(truel, probs) Area under the curve: 1   Could you please explain to me why it happens? No secondary button (right click) on touchpad I'm experiencing a serious problem with the new Ubuntu 18.04:  System Dell XPS 13 with a freshly installed Ubuntu 18.04 LTS.  Problem description The touchpad provides only primary button (single click and double click), but no secondary button. In particular, when using the "test Your Settings" option in Settings -> Devices -> Mouse &amp; Touchpad, any click on the touchpad is signaled as primary button, no matter where I click (left or right side of the touchpad). This remains the case even after I change the Primary Button from Left to Right (in Settings -> Devices -> Mouse &amp; Touchpad -> General).  In comparison, an external mouse connected to my laptop (via usb) works fine: it provides both primary and secondary buttons with sides corresponding to the choice of Primary Button (in Settings -> Devices -> Mouse &amp; Touchpad -> General).  This may be related to   Any ideas?  Update: Meanwhile, I found out that in addition to the solution proposed by EntropyWins, one can also control this touchpad behavior via the Tweaks tool (a.k.a. gnome-tweaks) under Keyboard &amp; Mouse -> Touchpad -> Mouse Click Emulation. In any case, as EntropyWins explains in his answer, this is not a bug. Determine if a set of tiles on a grid forms an enclosed shape How should I act when I see tiny "Have you tried this..?" type of answers? Create hexagonal mesh on 3D Parametric Plot and 3D spheres? First week of August in Scotland : What kind of clothes to pack? In a couple of weeks I will be going to Edinburgh, Scotland. I've been told that the weather is very unpredictable throughout the year. Can you suggest what I should pack as regards clothes? Why did Doc Brown choose the Delorean DMC-12 as his first time traveling vehicle? According to , the movie makers decided     The specific choice of vehicle was a  for the purposes   of it looking like an alien spaceship due to its characteristic   gullwing doors.   Is there a reason from within the storyline (comic books, cartoons, video games, deleted scenes) that explains why Emmett Brown choose that particular make and model? Why does subsurf modifier cause weird deformation at the end of a cylinder? My problem is simple. I think that the strange deformations on the top are there because there are so many vertices in one face. How do I fix this while maintaining nice topology?  Although I triangulated the top polygon, the problem still persists. What is the "maybe" caller id suggestion in the call log based on? I got a call from a phone number that is not stored in my address book, and in the call log, under the number, a name was suggested: "Maybe: X" (where X is the guessed name).  Where is this guess coming from? Direct observations of a black hole? I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it by methods of complex analysis. But I have not taken this course yet. Thanks for any help. Complement of maximal multiplicative set is a prime ideal Should I assign dwellers to the storage room? I've played Fallout Shelter for a while now and normally use dwellers with appropriate SPECIAL stats for each room (Strength in the reactor, Perception in the water supply, etc).  However, I have a question regarding Endurance. The only room which uses this is the storage room. It doesn't produce anything; it has no cooldown; it just does nothing...  Even if I just build and upgrade the warehouse, my stock capacity is increased. Is there any good reason for me to waste my dwellers by assigning them to storage rooms? How do we see? Where do the photons disappear? Find a plane that passes through a point and is perpendicular to 2 planes Origin —  Elementary Number Theory — Jones — p35 — Exercise 2.17 — For which primes $p$ is $p^2 + 2$ also prime? Only for $p = 3$. If $p \neq 3$ then $p = 3q ± 1$ for some integer $q$, so $p^2 + 2 = 9q^2 ± 6q + 3$ is divisible by $3$, and is therefore composite. (1) The key here looks like writing $p = 3q ± 1$. Where does this hail from? I know $3q - 1, 3q, 3q + 1$ are consecutive. $p$ is prime therefore $p \neq 3q$? (2) How can you prefigure $p = 3$ is the only solution? Does $p^2 + 2$ expose this? On an exam, I can't calculate $p^2 + 2$ for many primes $p$ with a computer — or make random conjectures. Right click not working on Ubuntu 18.04 I have installed Ubuntu 18.04 LTS on my HP Pavilion series laptop. After installing this version of Ubuntu, right click is not working of my touchpad. How can I solve this?  BTW:  does not have any solution which which is working for me. PS:- Problem remains after selection area in gnome-tweak-tool