Hello. I have a probably trivial commutative algebra question. If $R$ is any commutative ring, why do we have $\dim R[x]\geq \dim R+1$? When $R$ is not a domain the ideal $(x)\vartriangleleft R[x]$ may not be prime (take e.g $R=\mathbb Z/4$ and $x+2\in \mathbb Z/4[x]$).
may someone tell me if my proof on that every subset of a discrete metric space is open, is correct, please?
Case 1) If A is the empty set, then A is open.
Case 2) Let A be a non empty subset of the $X_{discrete}$ . Let a$\in$A and set r$=$ $\frac{1}{2}$ then if x$\in$ $N_r(a)$ then $d(x,a)=0$ and so $x=a$, hence x$\in$A.
Mathematics is the art of generalizing phenomenons and concepts. One of the most representative symbols in mathematics is the letter x, and it perfectly encapsulates the idea of generalization: we might observe the area of a square with side length 3 to be 3 times 3, and the area of a square with side length 5 to be 5 times 5, so we might extrapolate and claim that the area of any square is its side length times itself, or in short: the area of a square with side length x is x times x, demonstrating how the symbol x is related to generalization.
@mathsssislife why divide into two cases when your proof of case 2 did not rely on the non-emptiness?
the art of generalizing phenomenons and concepts -> the art of generalizing patterns
@LeakyNun what do you mean by that my proof of case 2 did not rely on non-emptiness? I assumed it is non empty,and since it is non empty it has an element.
@Leaky I was a math undergrad, I graduated in September 2018, was supposed to go to Heidelberg this year for my masters but deterioration in my mental health meant that I failed to get my application in on time l o l
My plan is thus; if my interview on tuesday goes well and I get the job then I'll apply in March 2020 (if I enjoy the job) so that I can start in the summer semester 2020
this means I can save money for the time, and do some more self-studying (I have no problem with this anyway, all of my pure maths knowledge is self-studied)
@Leaky at least you have pure stuff available to you, almost my entire curriculum was statistics and applied maths, both of which are entirely mystical to me (rendering the majority of my degree useless in my eyes, though not in the eyes of employers, but who cares about them?)
@LeakyNun so all I should have said was: Let A be a subset of $X_{discrete}$ . Let a$\in$A and set r$=$ $\frac{1}{2}$ then if x$\in$ $N_r(a)$ then $d(x,a)=0$ and so $x=a$, hence x$\in$A. am I right?
@Jacksoja cheers :) if I'm honest I don't follow it particularly well, but I identify completely as European rather than British, so it's like being torn from your country because some fat old racist white men decided it's the best idea
Excuse me if I cause offence (but also not if you voted leave)
@ÍgjøgnumMeg Let $G$ be a finite group, $K$ be a field, define $I_G := \{ \sum_{g \in G} \lambda_g g \mid \sum_{g \in G} \lambda_g = 0 \}$, show that $I_G$ is generated as a $K$-vector space by $\{g-1 \mid g \in G\}$
What is a homomorphism defined on the group of invertible upper-triangular $3\times 3$ matrices whose kernel consists of matrices $\begin{bmatrix} 1 & 0 & a \\ 0 & 1 & 0 \\ 0 &0 & 1\end{bmatrix}$?
I want to use this to study the quotient group, also is there always a way to find a group homomorp...
Guys, are these two statements equivalent: A set U is open if for every u$\in$ U $\exists$ $r>0$ such that $N_r(a)$ $\subset$ U , A set U is open if $\forall$ u$\in$U $\exists$ $r>0$ such that if for every x $\in$ $N_r(a)$ then x$\in$ U. Also, If I just the remove the $\textbf{for every}$ x $\in$ $N_r(a)$ does that still mean i'm universally quantifiying over x?
it's interesting, though I lack the tools to understand anything but statements lol
Kummer's proof of Fermat's last theorem fails for irregular primes because his proof depends on the fact that the class number of the $p$th cyclotomic field (for $p$ regular) is prime to $p$, classical Iwasawa theory looks at the behaviour of irregular $p$ and why Kummer's proof fails at those primes
if someone who doesn't know what a group is asks you what a vector space is and you answer by saying it's an abelian group together with an action blah blah then it's not going to help
@ÍgjøgnumMeg for a prime $p$, find the JNF of the $p$-by-$p$ matrix $\begin{bmatrix}0&0&0&\cdots&0&1 \\ 1&0&0&\cdots&0&0 \\ 0&1&0&\cdots&0&0 \\ 0&0&1&\cdots&0&0 \\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots \\ 0&0&0&\cdots&1&0 \end{bmatrix}$ over the field $\overline{\Bbb F_p}$
Guys so I posted this linear algebra question about two days ago; I got an answer but i asked back so i could get some clarification. The person who answered did not respond so I'm asking here. This is the question:
I will look for some reference for this: given a nonzero integer $n,$ there are infinitely many primes for which $n$ is a square (a quadratic residue). The two proofs i can think of use quadratic reciprocity, Dirichlet's result on primes in arithmetic progressions, or Chebotarev density.
https:...
Does anybody get why the prime can't divide the determinant? Is it because then we're going to have a zero eigenvalue?
I thought so but you don't really need the matrix to be invertible
My best guess is that if you have a 0 eigenvalue then you can't have $D^n = I_2$ no matter what n, where D is the diagonal matrix, but I'm not sure that's it
I couldn't understood the proof of bezout's identity (en.wikipedia.org/wiki/B%C3%A9zout%27s_identity#Proof). specifically the statements, "The set S is nonempty since it contains either a or –a". Someone please help me to made me understand.
what if we choose y=±1 and x=0, is choosing 0 and ±1 for x,y going to give lowest value?
Hi everyone, "local homology groups depend only on local topology provided that points are closed" - I was reading that and can see why it might not/can't? work if points are not closed, and created some examples on finite topologies for that - but I would like to see an example which is significant/practically important etc.
I have done both of these in my math courses, but without understanding what they actually are intuitively. I would be very much grateful if you could give me an intuitive explanation of them.
In one or two sentences, what is the basic idea behind discrete Fourier transform, Fourier analysis, fast Fourier transform, etc., etc.? Please, one or two sentences for each of these concepts. Each of these one-two sentences paragraph should summarise the corresponding concept.
I want to calculate the solution $p\in \mathbb{P}^4$ of the interpolation exercise $$p(0)=2 , \ p'(0)=3, \ p''(0)=1 \\ p(1)=2, \ p'(1)=0$$
For that we have to write a general polynomial of degree 4, $p(x)=ax^4+bx^3+cx^2+dx+e$ and then with the given coditions we have to solve a linear system. Is that correct or am I supposed to do something else?
I have been stuck with an eigen boundary value problem. Have already asked two variants of it on MSE. I keep getting trivial solutions to my found EVs. It would be really helpful if anyone could have a look into it
I am trying to find the eigenvalues of this Eigen BVP. $\mu$ is the eigenvalue parameter
$$
\lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \beta_h^2 F = 0
$$ wit BC(s) $F(0)=0,\frac{F''(0)}{F'(0)}=\beta_h,\frac{F''(1)}{F'(1)}=\beta_h$
For $\...
@ÍgjøgnumMeg The order of $U(49)$ is $\varphi(49)=42$, since, by definition of the group of units and of $\varphi$, that is how many elements of the group there are; however, there is $\varphi(42)(=12?)$ generators of $U(49)$ because we are given that the group is cyclic.
I hope that's right. I could be wrong.
Tag me if you'd like to correct/verify me, please :)
for example if you look at $n = 3$ and look at the elements you have $1, 3, 5, 7$ coprime to $8$ but $1^2 \equiv 3^2 \equiv 5^2 \equiv 7^2 \equiv 1 \bmod 8$ so you have $(\Bbb Z/(2^3))^\times \cong \Bbb Z/(2) \times \Bbb Z/(2)$
I know it will seem trivial, but why do we care about joint distributions? Of course, we may care about the joint distribution of two or more variables, because we may be looking for answers to questions like "What is the probability that today will rain and my friend will visit me", etc.
Anyway, I am not really asking a question which I've just answered. I didn't finish it. Why do we particularly care about joint distributions in the context of causality, if any one here knows what I am talking about?
So the subcategory of $\mathsf{Sch}/X$ of schemes $\pi:Y\to X$ where $\pi$ is affine is equivalent to the category of quasicoherent $\mathcal O_X$-algebras, is the full slice category $\mathsf{Sch}/X$ equivalent to a nice category of algebraic objects?
@TobiasKildetoft what was the question? (asking for a friend)
@ÍgjøgnumMeg isn't that just another way of saying that it splits as the product of $m$ non-zero rings?
the idempotent of any ring A corresponds to clopen subsets of Spec(A) anyway
@ÍgjøgnumMeg the trick is to "pull the $p$ out", i.e. 3rd isomorphism, i.e. $\Bbb Z[X]/(p,(X+a)(X+b)) = \Bbb F_p[X]/((X+\overline a)(X+\overline b))$ and $\Bbb Z[X]/(5,(X+1)(X^2+2)) = \Bbb F_5[X]/((X+1)(X^2+2)) = \Bbb F_5[X]/(X+1) \times \Bbb F_5[X]/(X^2+2)$
@LeakyNun (this is part 4, but I will just let you work out the entire thing with no helper-questions). Let $G = D_5\leq S_5$ and let $V$ be the permutation representation coming from the action on $\{1,2,3,4,5\}$. Decompose this as a sum of irreducibles and find the characters of the summands.
@Leaky well the proof I saw splits $(0)$ into $$\bigcap_{i = 1}^m \mathfrak{m}_i^k$$ so that $R \cong \prod_{i = 1}^m R/\mathfrak{m}_i^k$ and uses that each of the factors there is a local ring (wherein the only idempotents are $0$ and $1$)
I have done both of these in my math courses, but without understanding what they actually are intuitively. I would be very much grateful if you could give me an intuitive explanation of them.
