What is the inverse of X modulo $1 + X + X^2 + X^3 + X^4$? Is there any open softwares to calculate such things easily?
Let $\theta$ be a root of $p(x)=x^3+9x+6$, find the inverse of $1+\theta$ in $\mathbb{Q(\theta)}$.  So problems like this really annoy me but I did crappy on the last homework after making a lot of arithmetic mistakes so I want to run everything by you guys. This one isn't actually for homework it's just a suggested practice problem but okay lets go!!!  So I'm not sure if the method I used to do this was standard or not, I don't remember my professor showing me, but what I did first was used the canonical euclidean algorithm in $\mathbb{Q[x]}$ and wrote:  $x^3+9x+6$  $=(1+x)(x^2-x+10-\frac{4}{x+1})$  $=(1+x)(x^2-x+10)-4$  the reduced modulo  the minimal polynomial of $\theta$ i.e. $x^3+9x+6$ and so  $4=(1+x)(x^2-x+10)$  $\rightarrow$  $1= \frac{1}{4}(1+x)(x^2-x+10)$ and sooo ah $(\theta^2-\theta+10)\frac{1}{4}$ is what I believe is the inverse of $(1+\theta)$ in the quotient field... Like i said I hate these problems but ehh is this correct? On a lighter note it's finally getting cool enough for me to go on proper runs here and so that's pretty bomb!
Where is the precmd command in android? I'm kind of used to linux. I'm using Busybox from the Busybox Free installer, with Busybox 1.21.0-Stericson. Trying to chance my shell to the Busybox shell using busybox ash gives the following error after every single command:  ash: precmd: not found echo $PS1 $(precmd)$USER@$HOSTNAME:${PWD: -?} $    I tried to go back to the normal shell and find out WHERE precmd is:  which precmd   It just changes the prompt to have "1|" in front of it, without giving me WHERE it is. I found a solution , but have no idea where to put it. I'm basically trying to do the same as  (make Busybox shell the default). It is given in a question that $\vec x+\vec a=(\vec a\cdot \vec x)\vec b$ and it is stated that $\vec a$, $\vec b$ and $\vec x$ are vectors.  It is also stated that $\vec a\cdot \vec b\ne 1$.  The question asks to express $\vec x$ in term $\vec a$ and $\vec b$.  How do you this? Would anyone can help me how to show that a finitely generated projective module over a local ring and PID are free?    What I know about a finitely generated projective module $M$ over a PID $R$ is isomorphic to $R^k\oplus R/(a_1)\oplus\dots\oplus R/(a_n)$, and for the local ring case I don't know how to start. When can I make opportunity attacks? If Newton's third law is true, why can we sink in sand? Newton's third law of motion states that every action has an equal and opposite reaction. That is the reason we do not sink into the earth, because when our weight exerts a force on the earth it also exerts an equal and opposite force on us.  But when we stand on quicksand or on fluids we can sink in. How is this possible? Does it not exert an equal and opposite force on us? Or are Newton's laws different in the case of fluids and substances of low densities? Find $\int \frac{x^3}{x^2-1} \:\mathrm{d}x$ Find $\displaystyle \int \dfrac{x^3}{x^2-1} \:\mathrm{d}x$.   I think there are a few ways to do this. We could just break this up using long division or break it up into partial fractions then use long division. My homework says to break this up into its roots so that's why I brought up partial fractions. Is that the best way to do it here? Should I use exit() or just return statements in main()? Personally I favor the return statements because I feel it's like reading any other function and the flow control when I'm reading the code is smooth (in my opinion). And even if I want to refactor the main() function, having return seems like a better choice than exit().  Does exit() do anything special that return doesn't? I want to better understand why the construction something + [adjective] makes grammatical sense. Background: I was recently reading a Washington Post article and came across the following sentence.  So if you fall in that category, [the movie] feels more like magic than repetitive.  (Note: I've copied the relevant spoiler-free sentence above; following the link to the article will open you up to movie spoilers! ) The phrase more like magic than repetitive struck me as odd, presumably because it was non-parallel (more like [noun] than [adjective]). I would have expected &quot;...more like magic than something repetitive.&quot; (Actually, that needs to be rewritten to flow better, but the point is that at least it appears to be grammatical.) Thus, one would have the noun phrase &quot;something repetitive,&quot; of the form &quot;[noun] [adjective].&quot; This got me to thinking about the use of adjectives after nouns. There is a great Stack Exchange article  that involves the characteristics of the adjective that indicate it should come after the noun. But I haven't been able to find examples of how the word &quot;something&quot; requires a postponed adjective. Consider the following sentences:  I want to buy something red. I am looking for something sweet to eat.  You cannot say:  *I want to buy red something. *I am looking for sweet something to eat.  But you can say:  I want to buy something (that is) red. I am looking for something (that is) sweet to eat.  So, my question:  Do the above examples mean that the &quot;something [adjective]&quot; construction is an example of the ellipsis of &quot;that is&quot;?  Is there something special about the word &quot;something&quot; that requires the postponed adjective? Note that I am NOT looking for details about adjectives that are postponed; I'm interested in the word that precedes the adjective (either &quot;something&quot; or similar words).   By the way, I am not interested in the following constructions, which are different:  I want to paint the house red. I tried to build the brick house taller.  Thanks in advance! How to backup settings and list of installed packages If I want to make a backup of everything I have done since the fresh install of Ubuntu, what are the possible options? What all should I backup? I want to get all the settings that I changed, all the packages I installed, etc. Is there an official rule that the minimum damage can't be negative? A slightly humorous thought occurred to me (a recovering rules lawyer) after reading .     "None of the answers actually specify that 0 is the minimum damage."   Logically it wouldn't make sense for an attack to apply negative damage or healing, except in specific cases like hitting a flame elemental with a fire attack. Ruling that you could do negative damage would lead to an exploit to avoid costs of healing. As such I highly doubt any DM would make such a ruling. I definitely wouldn't except for the most absurd and humorous of games were failure is the expected result not success.  Another possibility (as suggested by Secespitus) is that negative damage would reflect back on the attacker. This could be seen as punching a strongman so hard (or in just the wrong way) as to injure your own hand.  This would also play well in a failure is the only option game, and if there is a rule suggesting that it would be an appropriate answer to this question as well.  Take the case of a person with less than 8 Strength, no levels in monk, and does not have a tavern brawler.  Would they do -1(1 - 2) damage on unarmed strikes? If so what would the result of the negative damage be?   "Healing" the target because such a puny hit actually raises their fighting spirits and makes them more capable of fighting Reflective Damage, the attacker actually takes 1 point of damage, representing injury to his hand or other damage   I am curious to know if there is an official rule stating that the minimum damage an attack can do is 0 or is there something in the damage calculation that I am missing that would otherwise prevent negative damage? Why is my USB drive showing corrupted data when plugged as an internal SATA drive? I have a 3TB Seagate Backup Plus Desktop USB 3.0 drive, which works fine when in its enclosure, but when I get it off its enclosure and directly plug it as an internal SATA drive, it's just not properly recognized (it works again when used in the enclosure). My systems’s motherboard is an , which has two SATA 6.0 Gb/s and four SATA 3.0 Gb/s ports.  When used as an internal SATA drive Windows 7 asks if I want to format the drive, as if it didn’t have a proper filesystem, and if I use the DISKMGMT.MSC tool (typing that in the start menu) I get completely wrong information about the drive.  It says the drive has 3 partitions (349,31GB of RAW data and two unassigned partitions of 1698,68GB and 746,52GB). This information is plain wrong, since the drive, when used in the USB enclosure, works as a single NTFS partition (2794,52GB). How come it’s shown as 3 partitions without filesystem when connected through SATA?!  Is Seagate using some proprietary way of storing the data when using the drive in its provided USB 3.0 enclosure? I didn’t use any kind of encryption. I just plugged the drive with USB 3.0 and started using it straight away, since it apparently came pre-formatted with an NTFS filesystem. This is a question that has received various answers. This questions seems simple, but it proves to be rather tricky.  So :  We flip a coin (H-T) independently $10$ times. Let's for the moment not care about whether it is fair or not.   How many $3$-head sequences are there?  Solution:  Some say that the answer is $\displaystyle\frac{10!}{7!3!}$, but this is the number of ways  by which $3$ heads can be arranged in $10$ positions. This answer , however, does not say anything about the remaining $7$ positions.   Each of these seven positions can be H or T. The total number of $7$-element sequences is $1\cdot1\cdot1\cdot2^7=2^7$. Then we have to multiply this by $\displaystyle\frac{10!}{7!3!}$ as the three heads can be anywhere. But I have found that this is not the correct answer. The correct is just $\displaystyle\frac{10!}{7!3!}$. So why not multiply by $2^7$? YA novel about dystopian future with non-human overlords and teenage heroes Is there a way so that I make make it when I left click/right click with a certain item it applies an effect? I know that you can apply attributes to wearing armor and things but I want it so when you use the item it applies an effect on your player.  For example, let's say I want something so that when I right click with a certain item (let's say an iron ingot that was given to me) I'll then have a jump boost for 10 seconds. So kinda like a potion in a way.  Is this possible with command blocks/commands/any mods? Did Anakin and Obi-Wan use the Force to protect themselves from the heat on Mustafar? While duelling that close to lava, I would suspect that they would get dangerously hot and dehydrated, and possibly burned. Did they use the Force to protect themselves in some way, for example lower their body temperatures and cool the air around them while duelling? How do I get the title of a .pdf document to appear as the page title? I'm trying to figure out the name of a movie I saw on TV as a kid, in the mid-to-late 90s, in the US, in English, on daytime TV.  I remember it being kind of Starship Troopers-esque, but everyone had these glowing lights in their chests.  They glowed in different colored patterns, and I think it had something to do with whether you were attracted to someone or not (I was probably like 7 when I saw it, I didn't really understand the context of that part yet). Interscholastic Mathematics League Senior B Division #2 Points $P$, $Q$, $R$ and $S$ are chosen on the sides of parallelogram $ABCD$, so that $P$ is on line $AB$, $Q$ is on line $BC$, $R$ is on line $CD$, $S$ is on line $DA$, and $AP=BQ=CR=DS=\frac{1}{3}(AB)$. Compute the ratio of the area of $PQRS$ to the area of $ABCD$. Is it (if at all) possible in Blender 2.6+?   If not, will future versions of Blender support it?  I see some old posts ( and ) saying that it is not possible. However, I found  referring to operators under  that allowed me to add sensors and controllers from python.  bpy.ops.logic.sensor_add(type='ALWAYS', object=my_obj.name) bpy.ops.logic.controller_add(type='LOGIC_AND', object=my_obj.name)   However I am not able to find any way to link them from python. I tried following (but did not work):  new_controller = my_obj.controllers[-1] new_sensor = my_obj.sensors[-1]   It throws AttributeError saying that object has no attribute 'controllers' :-/ birthday problem - expected number of collisions There are many descriptions of the "birthday problem" on this site — the problem of finding the probability that in a group of $n$ people there will be any (= at least 2) sharing a birthday.  I am wondering how to find instead the expected number of people sharing a birthday in a group of $n$ people. I remember that expectation means the weighted sum of the probabilities of each outcome:  $$E[X]=\sum_{i=0}^{n-1}x_ip_i$$  And here $x$ must mean the number of collisions involving $i+1$ people, which is $n\choose i$. All $n$ people born on different days means no collisions, $i=0$; two people born on the same day means $n$ collisions, $i=1$; all $n$ people born on the same day means $n$ collisions, $i=n-1$.  Since the probabilities of three or more people with the same birthday are vanishingly small compared to two people with the same birthday, and decreases faster than $x$ increases, is it correct to say that this expectation can be approximated by  $$E[X]\approx {n\choose 0}p_{no\ collisions}+{n\choose 1}p_{one\ collision}$$  This doesn't look right to me and I'd appreciate some guidance.    Sorry - edited to change ${n\choose 1}$ to ${n\choose 0}$ in second equation. Sloppy of me.