An assignment problem( in elementary number theory course) on which i am struck and need help
For what powers $k$ is the polynomial $n^k-1$ divisible by $(n-1)^2$?
Filling a conical tank I have been working on this problem for about 2 hours and I can't seem to get it, here is exactly what the question reads. "Water is poured into the top of a conical tank at the constant rate of 1 cubic inch per second and flows out of an opening at the3 bottom at a rate of .5 cubic inches per second. The tank has a height of 4 inches, and a radius of 2 inches at the top. How fast is the water level changing when the water is 2 inches high?" SSL certificate for a public IP address? What word (or how do you phrase things) do you use when the ostensible word for the class is the same as the word for a subset of the class?  For example, in the United States, there are many brands of sugared, carbonated soft drink that one can buy: Coca-Cola (or Coke), Pepsi, RC Cola, Sprite, 7-Up, A&amp;W or Mug Root Beer, etc). The first two are kinds of 'coke' or 'cola' (but no one says 'cola').  , one asks for any type of such drink as 'coke'. How does one then ask for the particular kind of 'coke' that is 'Coke' (pronounced the same)? Does one reduplicate and say a 'Coke coke', does one repeat with emphasis (or without), does one use a different word like 'coca-cola' (co-cola), or what?  The (perceived) difficulty (surely those in the South are able to get the drink they want somehow) isn't limited to there. Where the drink is called 'soda', if you ask for a 'coke', you'll still sometimes have the choice among Coca-Cola, Pepsi, etc. "Can I have a Coke?" the response might be "Is Pepsi OK?". "Sure, Pepsi, Coke, whatever." You don't use 'cola', you just ask for the particular kind and hope the listener will help differentiate.   Given these multiple strategies for disambiguating, what do you do say to get a 'Coke' in the southern US? are there any other examples of a hypernym being the same as the hyponym, and so creating a disambiguation problem? How to install brackets.io from the command line? For some reason the gnome installer that handles .deb packages on my system is not working. I'm not concerned.  How would one install the Brackets.io program from the terminal? How to get client's IP address using JavaScript? I need to somehow retrieve the client's IP address using JavaScript; no server side code, not even SSI.  However, I'm not against using a free 3rd party script/service. string str1 = "12345ABC...\\...ABC100000";   // Hypothetically huge string of 100000 + Unicode Chars  str1 = str1.Replace("1", string.Empty);  str1 = str1.Replace("22", string.Empty);  str1 = str1.Replace("656", string.Empty);  str1 = str1.Replace("77ABC", string.Empty);   // ...  this replace anti-pattern might happen with upto 50 consecutive lines of code.   str1 = str1.Replace("ABCDEFGHIJD", string.Empty);   I have inherited some code that does the same as the snippet above. It takes a huge string and replaces (removes) constant smaller strings from the large string.  I believe this is a very memory intensive process given that new large immutable strings are being allocated in memory for each replace, awaiting death via the GC.  1. What is the fastest way of replacing these values, ignoring memory concerns?  2. What is the most memory efficient way of achieving the same result?  I am hoping that these are the same answer!  Practical solutions that fit somewhere in between these goals are also appreciated.  Assumptions:    All replacements are constant and known in advance Underlying characters do contain some unicode [non-ascii] chars Prove or disprove: $S_{10} = \langle (1,3),(1,2, ... ,10) \rangle$ On my y510p, I set Lenovos battery software to "Optimize battery health" while I was in Windows 8.1. I forgot to change it before overwriting that partition with Ubuntu, and now my battery in Ubuntu never charges past 59~60%. I think this is due to something in the BIOS because it persists despite my Windows partition being gone.   I've searched the BIOS but there's no mention of battery anywhere!  I'm using Ubuntu 14.04 LTS. Splitting of prime ideals in algebraic extensions I'm reading Algebraic Number Theory by Jurgen Neukirch. I have some problems with some of the exercises in Section 9 of Chapter 1.  They are:  1) If $L / K$ is a Galois extension of algebraic number fields with non-cyclic Galois group, then there are at most finitely many non-split prime ideals of $K$.  2) Let $L / K$ be a finite (not necessarily Galois) extension of algebraic number fields and $N / K$ the normal closure of $L / K$. Show that a prime ideal $p$ of $K$ is totally split in $L$ if and only if it is totally split in $N$.  I have worked on them for a long time but couldn't get any idea of it.   Can you please help? Thank you! I want to make an arena where mobs will fight each other and a player can bet on one side or the other. For example: There are 2 skeletons in dyed armor; red and blue. Each color representing a team. When they spawn I want them to fight each and then when one dies I want it to announce which team is the winner and give the player the "money" due. I'm just not sure how to get the skeletons to attack each other without a player instigating it. Number of Homomorphisms from $\Bbb{Z}_m$ to $\Bbb{Z}_n$ Where Android apps store data? Could you list all possible directories where Android apps may store data, providing description what kind of data are stored in each directory? My understanding is that a random variable is actually a function $X: \Omega \to T$, where $\Omega$ is the sample space of some random experiment and $T$ is the set from which the possible values of the random variable are taken.  Regarding the set of values that the random variable can actually take, it is the image of the function $X$. If the image is finite, then $X$ must be a discrete random variable. However, if it is an infinite set, then $X$ may or may not be a continuous random variable. Whether it is depends on whether the image is countable or not. If it is countable, then $X$ is a discrete random variable; whereas if it is not, then $X$ is  continuous.  Assuming that my understanding is correct, why does the fact that the image is uncountable imply that $Pr(X = x) = 0$.  I would have thought that the fact that the image is infinite, regardless of whether it is countable or not, would already imply that $Pr(X = x) = 0$ since if it is infinite, then the domain $\Omega$ must also be infinite, and therefore  $$Pr(X = x) = \frac{\text{# favorable outcomes}}{\text{# possible outcomes}} = \frac{\text{# outcomes of the experiment where X = x}}{|\Omega|} = \frac{\text{# outcomes of the experiment where X = x}}{\infty} = 0$$  What is wrong with my argument?  Why does the probability that a continuous random variable takes on a specific value actually equal zero? As it is common with people from my country, I have an immense difficulty with prepositions in English, especially with the use of in and on.   When the preposition indicates the position of the object it is a bit easier to decide which one to use, but as it requires thinking from me, I still say things like "I put the bill in the fridge", "my pencil is on my bag", etc, even though if I think about it, I would know how to say it correctly.  Of course, even in those cases where the preposition indicates position the decision is not always clear, as is the case for example with "on the bus" (you are not on top of the bus, you are inside the bus) or "he is in bed" (you are not inside the bed, you are on top of your bed).  Other cases are even less clear, and I believe there is no real rule to decide what should be said; the person should just know it. Two examples are "in a bad mood", "you are on my mind".  So my first question is whether there is really no rule to help me. The second question refers to two of the cases where I still don't know whether to use in or on:   good luck on/in your exam in/on the list   The last one is especially important to me, as I have asked a number of native speakers, and each time I get a different answer. The only consensus seems to be the case where you say "on top of the list". Apart from that, some people will say that I should say "his name is in the list" whereas others think I should say "his name is on the list". And yeah, a couple of people did tell me that I can use either.  Can someone help me? Original file claudio antonio claudio michele  I  want to change only the first occurrence of &quot;claudio&quot; with &quot;claudia&quot; so that I would get the following: claudia antonio claudio michele  I have tried the following: sed -e '1,/claudio/s/claudio/claudia/' nomi  The above command performs global substitution (it replaces all occurrences of 'claudio') . Why? I want to create a 2D game but I'd like to have the character to wear tons of different equipments on different combinations, like a RPG or something.  So let's say the user changes his/her shoulders and pants, this must be shown to the player and all these different equipments must follow the different characters action (hitting, being hit, spelling magic, etc.).  I see a few problems that I'd like to know the best approach/algorithm/architecture to solve.  1 - Sprites or Animations  Should I create different sprites for each equipment in each action animation?  Is it better to just have the equipment sprite and animate it directly on code like rotating and translating (using something like tween on Flash)?  Is there any other better option? (I really don't like the ideas above)  2 - Positions  Let's say during one of the character's move he starts looking directly in front to the player's view but ends his movement showing his right side to the player (like if he swings his sword from one side to the other).  If we consider one part of the user, like his head for example, it starts in front and then turns left.  It means at least 3 different positions for each head equipment (helmet, cap, whatever).  This surely influence the answer of #1 question. How is the best way to achieve that?  3 - Layers  Let's consider a character move where he spins 360 degrees with his arms open. At the beginning of the animation, his right hand is close to the user's view and then, at the middle of the movement, this is probably behind the character's body on the animation.  Whatever is the option on #1 question, you surely needs to use some kind of layer model to make this sprite or animation to starts closer to the player's view and later change to far away from player's view.  Is there any good way to do that?  I know the question is quite long and difficult to understand. Let me know if you think it's better to have some drawings to try to explain which one and I'll try to do that. How it this possible? Even if the gif is fake, the Mythbusters did it and with a large sail it really moves forward. What is the explanation? Why can the entropy of an isolated system increase? Anthology series with suspended animation and a galaxy-wide empire? Is the orbit in Schwarzshild metric a path with maximal proper time? In curved spacetime particles follow timelike geodesics, which should have maximal proper time (at least locally). I thought this path usually corresponds to a global maximum, and there are only strange exceptions. Propably I was wrong, or I really don't understand something, because it doesn't seem to be true for the case of orbitals in Schwarzschild geometry.  The Schwarzschild metric is (with +--- signature)  $$ g_{\mu\nu} = \textrm{diag}\left(1-\frac{2GM}{r},-\left(1-\frac{2GM}{r}\right)^{-1},-r^2,-r^2\sin^2\theta\right) $$  An observer with fixed $r,\theta,\phi$ coordinates will age according to $d\tau_1 = \sqrt{g_{tt}}dt$. An observer, which orbits the center at the same constant radius with $\theta=\pi/2$ will age according to $d\tau_2 = \sqrt{g_{tt}-\omega^2r^2\sin^2\theta}dt$, so he ages slower during the same amount of coordinate time $dt$. If the two observers start from the same point, they will meet again after a full rotation of the orbiting observer, and the orbiting observer will age less. What I don't understand is that the orbit is the geodesic (if the angular velocity is set properly), so why didn't he age more? He should have been on the path with maximal proper time.