This has to do with dimension; a PID is 1-dimensional, if you quotient by substantial stuff it becomes 0-dimensional - indeed, $R/I = R/(f_1^{r_1} \cdots f_n^{r_n}) \cong R/(f_1^{r_1}) \times \cdots \times R/(f_n^{r_n})$ is semilocal
the Euclidean norm just comes from writing an element as $(unit) f_1^{k_1} \cdots f_n^{k_n}$, $0 \leq k_i < r_i$ and then looking at $k_1 + \cdots + k_n$
What's the relationship between Krull dim of R and topological dim of Spec R? (Not sure what topological dim is appropriate here, probably the small inductive dimension since Specs are ugly)
Good afternoon everybody. Just for your comment. Is it clear my question now in English language and for my request? Thank you very much for all users into chat.
Supposing to have these integrals, for example,
$$\int\limits_{-\infty}^{+\infty}\frac{dx}{(x^2+1)^2} \tag 1$$
$$\int\limits_{0}^{+\infty}\frac{x^2}{x^4+1}dx \tag 2$$
$$\int\limits_{0}^{2\pi}\frac{1}{2\cos x+5}dx \tag 3$$
these integrals can probably be solved in the traditional way.
What if are...
@Alessandro Let $G$ be a graph, $E$ be the set of edges. Choose some distribution/measure $\mu$ on $\Bbb R_+$. For each edge $e \in E$ assign a random variable $X_e$ to it which follows the law of $\mu$, and $\{X_e\}_{e \in E}$ are all independent
In other words, consider the product measure $\mu^{\otimes |E|}$ on $\Bbb R_+^E$
The random variables $X_e$ are to be thought of assigning random lengths to the edges $e \in E$. Thus, we get a "random metric" on $G$ by declaring $d(v, w) = \inf \sum_{e \in \gamma} X_e$ where infimum varies over all edge-paths $\gamma$ on $G$ starting at $v$ and ending at $w$.
$d(v, w)$ itself is a random variable, of course, defined on $(\Bbb R_+^E, \mathcal{B}_{\Bbb R_+}^{\otimes |E|}, \mu^{\otimes |E|})$
In less idiosyncratic circles, this is known as first passage percolation :P
Thorgott is there any chance you can take a look at a theorem in a paper for me and tell me if you think it's a useful extension to the coarea formula for my purposes or not
Yeah, I initially made a comment noting that it's certainly not a smooth manifold, but at best one with boundary and that still wouldn't be smooth, because it's boundary would necessarily be the unit square, which is not a smooth submanifold, but then I noticed that even the definitions are messed up
@Drathora that would depend on how restrictive being in $W_{1,n}^1$ is and I have no clue what that is (some kind of Sobolev space, I wager, but not the usual ones)
we're proving the Hodge decomposition: If $(M,g)$ is a compact Riemannian manifold, you have $\Omega^k(M)=d\Omega^{k-1}(M)\oplus\delta\Omega^{k+1}(M)\oplus\mathcal{H}^k(M)$ orthogonally and $\mathcal{H}^k(M)$ is finite-dimensional, where $\mathcal{H}^k(M)$ is the space of harmonic $k$-forms
and in order to do this we assume elliptic regularity on manifolds and Rellichs compactness theorem without proof
and I'm feeling very lost
we spent like one page proving the Laplace operator is reversely bounded on the orthogonal complement of the harmonic forms
somehow this will ultimately imply that each cohomology class possesses a unique harmonic representative
