also if i understand correct, if $a_{\infty}$ converges to some finite value, then $\sum_k^{\infty} (-1)^k.a_n$ will always oscillate between two values right?
Spending the next couple of months trying to refresh my complex analysis so that should be enough to feel ready (although my rapidly deteriorating mental health has hamstrung me into total unproductiveness)
If we look at the series $\sum_{k=1}^\infty {}^k n$ for $n \in \Bbb Z$ in the p-adics, then it's clear that it converges pretty rapidly whenever $p$ divides $n$. I wonder if one can say anything about the limit
For a filtration $F = \{F_n\}$ ($n \in \mathbb{Z}$) on a $R$-module $A$ to be convergent we need their intersection to be zero, but we don't necessary need $F_i \cap F_j = 0$ for $i \neq j$ do we?
so the idea is: if you have some set $X$, then what should an abstract notion of "distance" on $X$ satisfy? The conditions we put on a function $d:X \times X \to \Bbb R$ (this means that we take two inputs from $X$ and get a real number) are really intuitive: - for all $x,y \in X$, $d(x,y) \geq 0$, because distance is always positive - for all $x,y \in X$, $d(x,y)=0$, because the only point having distance zero from $x$ should be $x$ itself - for all $x,y \in X$, $d(x,y)=d(y,x)$, because distance from $x$ to $y$ should be the same as the distance from $y$ to $x$
@Mathphile number theory is nice for non-experts, because there are plenty of elementary things that are fascinating to explore. But if you want to go "deeper", then you need other parts of math as well, such as algebra or analysis
I searched for primes of the form $p^p+2$, where $p$ is prime for a range of $p \le 10^5$ on PARI/GP and found that 29 is the only prime of this form in this range.
Questions:
$(1)$ Is $29$ the only prime of the form $p^p+2$, where $p$ is prime?
$(2)$ If not, then are there a finite nu...
Using PARI/GP, I searched for primes of the form $n!\pm k$ where $k \ne 2$ is prime and $n\in \Bbb{N}$.
With the help of user Peter, we covered a range of $k \le 10^7$ and couldn't find a prime $k$ for which $n!\pm k$ has no primes.
Observations:
$(1)$ When $n \ge k$, $n! \pm k$ cannot be p...
@Ryan rofl, nah this is an English university that focusses mostly on statistics and applied mathematics, so they didn't care much for rigor. Most of the courses I took were just plug and chug calculus courses
@anakhro I like the taste, and it's a habit I guess, I'll probably stop once I reach Germany but atm I'm such a mess that I have no willpower to stop anything hahaha
From my limited understanding, strictly positive elements in a $C^*$-algebra $\mathcal{A}$ can be defined in one of two (equivalent) ways:
(1) $x \in \mathcal{A}_+$ is strictly positive provided $\overline{x \mathcal{A}x} = \mathcal{A}$
(2) $x^\ast = x$ is strictly positive provided $\ph...
Suppose you have a sequence, such as $n^4\text{ mod }(4n^3-4n^2+n-1), n\in\Bbb N$. Are there methods to analyse the sequence and measure how "random" the values are?
no i mean that $\lim_{n \to \infty} \sum_{k=1}^{n} (-1)^k(^kx)$ oscillates between two different values depending on whether $n$ is odd and even only for $e^{-e} \lt x \le e^{1/e}$
ok if C(X) is Jacobson then every prime is maximal so Spec(C(X))=mSpec(C(X))=X (Ex.1.26). Spec(C(X)) is Hausdorff so Spec(C(X)) is totally disconnected (Ex.3.11). So X is totally disconnected.
so to prove the claim about sober spaces, note that if X is a top. Space and Y is a very dense subspace, then the map Ouv(X)->Ouv(Y), U->U \cap Y is an isomorphism of locales (or of sites, if you want)
but for sober spaces, the embedding into locales X->Ouv(X) is fully faithful
so that proves in our case since both Specm(C(X)) and Spec(C(X)) are sober, both are homeomorphic, so Spec(C(X)) is Hausdorff and thus totally disconnected and so is X=Specm(C(X))
@LeakyNun it's an isomorphism of sites, really. If Y is very dense in X, then you can recover each open subset from its intersection with X: this is clear for closed subsets, since by definition of being very dense $\overline{Y \cap A}=A$ for any closed subset. So by taking complements, this is also true for opens
From my limited understanding, strictly positive elements in a $C^*$-algebra $\mathcal{A}$ can be defined in one of two (equivalent) ways:
(1) $x \in \mathcal{A}_+$ is strictly positive provided $\overline{x \mathcal{A}x} = \mathcal{A}$
(2) $x^\ast = x$ is strictly positive provided $\ph...
