i had a big problem earlier this semester with students distributing exponents across addition (i.e. $(a+b)^2=a^2+b^2$) to the point where i had to issue an outright ban on it, zero points on the entire problem even if everything else is right
@FernandoMartin The area of a semicircular sector given by an angle of $\theta$ radians is $x=\cos \theta$, $$\frac{x\sqrt{1-x^2}}2+\int_x^1\sqrt{1-t^2}dt$$
okay here is where you can find the answer: Corner, A. L. S. On endomorphism rings of primary abelian groups. Quart. J. Math. Oxford Ser. (2) 20 1969 277–296
@FernandoMartin It is actually Krull-Schmidt. It says that a group that if both Noetherian and Artinian can be decomposed into a product finitely many idecomposable groups.
If you have $G=G_1\times G_2\times\cdots\times G_s=H_1\times\cdots\times H_t$, all factors indecomposable, we can reindexing so that $G_i\simeq H_i$ we can write $G=G_1\times \cdots \times G_l\times H_{l+1}\times \cdots H_t$.
@ThomasAndrews I keep asking for comments on related material and no one seems to get it. I've formalized and formalized and that just makes things worse. I don't know how to make it clearer if the community doesn't speak.
He's being persecuted! He's halfway on the way to being a triangulator, I suspect.
user4704
Please keep it civil.
user4704
Downvotes without commentary are an unfortunate way of life on StackExchange. Users are not required to explain their down votes... for better or worse... and it's best to try and not let it get to you on a personal level.
anyway @Enjoys, I can send you some tips on how to make your questions more well-received if you like. people can be pretentious and impatient here, but there are tricks you can use to make your questions more likely to be answered.
I suspect the downvoters (I am not one of them) are doing so because your question is a definition (a very unclear one, from my reading,) and then you ask, "Is this new? Is this useful?" Those kinds of questions are generally discouraged. We can define lots of things in mathematics, so if you have no motivation for defining that thing, then it's unclear why we'd care. It's more about, "Have I invented something new?" @EnjoysMath
I've been getting a lot of downvotes on old questions in the last few days. One question got downvoted twice, so it wasn't just one person, unless he was really determined...
@PedroTamaroff Somebody downvoted all the 1 hour or older answers to that question, except for one. I think we have a suspect.
Oh I deleted it, but it's $s^+ (abc + def) = def + abc$ the operator on the left is part of a group acting on all related expressions. Where related expression means it can be gotten to by using your ring axioms.
One confusion for me was the question said $abc+bde + xyz$. The repeated $b$ seemed a problem, because you couldn't then explain the question as a permutation of variables. @EnjoysMath @AlexanderGruber
@EnjoysMath what i'm saying, though, is that $def + abc = abc + def$, so $s^+$ acts by the identity on any monomial expression ($s^+(abc+def)=abc+def$)
Two expressions can be equivalent on their polynomial maps, but if you try to compute using one over the other, one could lead to more computations, if by "computing using" you take to mean reading the expression directly and performing all seen operations.
@AlexanderGruber $a + b$ as an expression, there's a map from expressions to their underlying map. I just use the same notation because that's what we're used to working with. What else would I use to represent a polynomial expression other than the expression itself?
Yeah, that was the word that was heavily non-standard. It still strikes me as deeply uninteresting - the actions that you listed never reduced computations. @EnjoysMath
The set of "expressions" is just a set of binary trees with some labeling - leaves are either constants from $R$ or variables, and the other nodes are selected from the binary operations, $+$ and $\cdot$ here. It's still not clear what set the group action acts on.
Except you haven't actually defined that action. For example, if $g$ sends $1+x+y+xy$ to $1+x+y+yx$, what does $g$ send $(1+x)(1+y)$ to? You have to define the action on all the elements of your set.
Okay, well, you haven't defined a group action on the set, nor have you defined a group, and I'm not sure you have given me any reason to trust your gut.
You certainly have an equivalence relation on expression trees.
That's the risk. Did you really think you'd change the world?
I was young once. I tried to prove Fermat's Last. The question is, can you have the humility to recognize that a lot of very very very smart people have come before you and banged their heads on problems to no avail. It doesn't mean you shouldn't try, but in trying, keep some perspective.
One problem I see is that $(x_1 + 1)\cdots(x_k + 1) + x_1 x_2 x_3$ could be a smallest expression for that polynomial but unreachable unlesss I include the $a + (-a) = 0$ axiom.
Perhaps, but not everybody "invents group theory." That's the thing, you have delusions of grandeur well before you have actually accomplished anything. Galois invented groups and solved a problem with them. There was motivation. Here, you are talking about a formal manipulation that is trivially part of the question of evaluating polynomials, and tons of people have investigated those operations.
The formalization you are talking about is essentially trivial. It's certainly possible that something could pop out of it, but it seems highly unlikely. Galois was not formalizing something trivial when he invented groups - he found groups, and then formalized.
Hi all I have posted my question here http://math.stackexchange.com/questions/755241/set-geometry-and-inclusion and nobody has answered or even commented could you please have a look and tell me if it's at least clear
Actually I mainly want to compute the distance between a center of a circular set and the furthest edge of a bigger set that contains the first one
i've read through it like 3 or 4 times now and i still can't really tell what it's about
there's just too many symbols and definitions all crammed together, people aren't answering it because by the time they get halfway through reading it they decide it isn't worth the effort to sort out
put some story behind it - why is this particular problem interesting, where does it come from, which parts are which
@AlexanderGruber Well at least I've drawn attention to the issue. Could you perhaps unlock the post tomorrow in case anyone else wanted to answer? I will not use it for discussion, I will use the chat room now that I know it exists.
