@SirCumference It is consistent with $\mathsf{ZF}$ that $\Bbb R$ is a countable union of countable sets! It is also consistent with $\mathsf{ZF}$ that $\mathrm{cof}(\omega_1)=\omega$
and... you cannot construct a computable example of such $\Bbb{R}$
While there already exists a question regarding the clarification of the meaning of "conceptual inverses" and an excellent answer to it, I am failing to understand what exactly is the concept with respect to which the adjoint functors can be considered as "conceptual invesers" of each other.
Can we analyze the function $f(z) = \displaystyle \prod_{k=0}^\infty \left(1-z^{2^k}\right)$?
I know that the radius of convergence is $1$
where exactly does this power series converge?
It converges to 0 trivially whenever $z \in \Bbb C^\times$ is $2$-torsion, but I don't know about the other points
the 2-torsion points is dense in the unit circle btw
and if we do analytic extension, where do we get poles?
We also get interesting functional equations such as $f(-z) = f(z) \left( \dfrac{1+z}{1-z} \right)$ and $f(iz) = f(z) \left( \dfrac{(1-iz)(1+z^2)}{(1-z)(1-z^2)} \right)$
they might be helpful
I think $\omega$ is a pole, but I doubt whether it can be extended beyond the unit circle at all
if $\{z \mid \exists n, z^{2^n} = \omega\}$ is dense on the unit circle then I have a proof that it can't be extended beyond the unit circle
1 is definitely a limit point
this set is invariant under multiplication by the 2-torsion elements
therefore by translation every 2-torsion root of unity is a limit point
therefore the whole circle consists of limit points
therefore it can't be extended beyond the unit circle
So I wrote this way back in 2014. In the five years since, I have completely forgotten how I found this equation, or why it's true. So, um, if anyone could rediscover my proof, I'd be pretty grateful. — Akiva Weinberger1 min ago
Lol, when I wrote that, I just kinda assumed the reader would know what the digamma function is
For integer arguments, it's $H_{n-1}-\gamma$, I think, where $H_n=1+\frac12+\dotsb+\frac1n$
I think it's the log-derivative of the Gamma function (so $\psi=(\ln\Gamma)'=\Gamma'/\Gamma$)
@AkivaWeinberger As far as how one might prove your version, I agree with commentators that Gauss's digamma formula seems like the obvious approach
Main thing I'll add in that vein is that, in your formula, it isn't immediately obvious that $\psi(a/b)$ ought to be real-valued
But $\psi(x)$ is in fact real on the real axis. So I suspect that, if you take the real part of your formula and simplify a bit, you'll get the Gauss formula
So, I'm trying to set up an exhaustive computer search for a very large space (currently, there are $(n-1)!(n-1)^{(n-2)^2}$ cases to check, but I'm trying to narrow it down further). Would it be better to set up something in MATLAB or mathematica or set up an executable via C or some other higher level language?
@LeakyNun I'm searching the space of multiplication tables for magmas to find ones following a particular set of rules. I'm representing the table as an $n\times n$ array of arrays.
"The digamma function is often denoted as $\psi_0(x)$, $\psi^{(0)}(x)$ or Ϝ (the uppercase form of the archaic Greek consonant digamma meaning double-gamma)."
Well, there are two. First, for distinct elements $r,s$ we have that $ra=ar=s$ for all $a$ in the magma (easily controllable in the table, so I don't need to check for that). The other is $(ab)c=a(bc)\iff a=c$
My question is on the notation of the Digamma function.
The Factorial function $n!$ (which is met in secondary school), is conceptually seminal to the Digamma function. The Factorial function is defined as:
$$0!=1,\qquad n!=\frac{(n+1)!}{n+1}$$
This concept is extended with Gauss's Pi fu...
At one point I was trying to figure out a way to get glagolitic script into latex so I could use it for variable names. (solely for my own entertainment)
In mathematics, the trigamma function, denoted ψ1(z), is the second of the polygamma functions, and is defined by
ψ
1
(
z
)
=
d
2
d
z
2
ln
Γ
(
z
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{\displaystyle...
And the reason it's $1/2$ for the odds that it's less than a half is that if we reflect it ($\eta_n\mapsto1-\eta_n$) the sum becomes 1 minus the original
So then $e^{i t X}=\exp(i t \sum_{n=1}^\infty \eta_n 2^{-n})=\prod_{n=1}^\infty \exp(i t \eta_n 2^{-n})$
And since the $\eta_n$ are all independent, the expected value is simply $E[e^{it X}]=\prod_{n=1}^\infty E[e^{i t \eta_n 2^{-n}}]$
I guess I'm commiting a sin here
I shouldn't be using $\eta_n$ for the value of the random variable and for the random variable itself
ignoring that, though: since those are uniform random variables, we have $$E[e^{i t\eta_n 2^{-n}}]=\int_0^1 e^{i t \eta 2^{-n}}\,d\eta = \frac{e^{i t 2^{-n}}-1}{i t 2^{-n}}$$
Here's a problem which seems pretty tough, inspired by my running some statistics on $X=\sum_{n=1}^\infty \eta_n 2^{-n}$
(well, strictly speaking on $X^{(m)}\equiv \sum_{n=1}^m \eta_n 2^{-n}$, since I can't have mathematica generate infinitely many uniform random samples)
What's the equivalent of the central limit theorem for $X?$
in other words, suppose I take $N$ samples $X_1,X_2,\cdots,X_N$ of $X$
One thing about continuous functions is that if $x_n\to x$, then $f(x_n) \to f(x)$, so you can say give me any real number $a$, choose a sequence of rational numbers $a_n$ converging to $a$, then $f(a) = \lim f(a_n) = \lim 0 = 0$
Ransom question: Do you think one gets better at constructing mathematical arguments with excessive details and reasoning your way up from the assumptions in your problem, or is it accumulating a lot of "mini-propositions" in your head that let you operate at a "higher-level" ?
This occurs to me whenever I encounter the phrase "according to the non-trivial theorem..."
I feel like as you get used to really basic stuff you begin to reference it second nature
You begin to say things like "By property X, we can do Y" when the fact that property X allows you to do Y is something that a few weeks ago would've been a non-trivial theorem to prove
But is now a fact that you can state very inexplicitly
I have a habit of obsessing over the basics for a long time, which holds me back from that... I'm afraid I might do some error and my memory is kind of weak
If can prove to myself everything I use then the probability of doing some faulty step becomes slim
But then I will need a long-long time to familiarize myself with everything I learn
So, if you actually are forgetful and liable to either not remember theorems you've already proven and reprove, or somehow think something's right when it isn't, then I guess it's something you'll need to contend with/work around. It's definitely gonna be an added difficulty
I am enjoying mirror symmetry. I now know that I would like to put my effort into mirror symmetry, algebraic cycles, and Hodge theory for my next few years
@Daminark it's going pretty well - other than things that im supposed to be reading, ive been trying to learn a little bit of probability theory for fun
Lmao, yeah part of what I liked about my measure theory class was that we skipped certain proofs our professor felt weren't enlightening (e.g. Lebesgue measure is Radon)
Define $S = \{ s^2 : s \in \Bbb{Z}\}$ be the squares in $\Bbb{Z}$. Let $\chi_A(x) = $ the characteristic function of set $A \subset \Bbb{Z}$ which is another way of saying it's $0$ except when $x \in A$, then it's $1$.
For ideals $(m), (n)$ in $\Bbb{Z}$ these functions have the basic property $...
I need to solve cos(z)=cosh(z) , z any complex number. I've reached to a system of equations { sinxsinhy=-sinysinhx , cosxcoshy=cosycoshx } , but I don't know how to proceed. Any hint ? Or any alternative solution ? . Thanks
My question is on the notation of the Digamma function.
The Factorial function $n!$ (which is met in secondary school), is conceptually seminal to the Digamma function. The Factorial function is defined as:
$$0!=1,\qquad n!=\frac{(n+1)!}{n+1}$$
This concept is extended with Gauss's Pi fu...
Define $S = \{ s^2 : s \in \Bbb{Z}\}$ be the squares in $\Bbb{Z}$. Let $\chi_A(x) = $ the characteristic function of set $A \subset \Bbb{Z}$ which is another way of saying it's $0$ except when $x \in A$, then it's $1$.
For ideals $(m), (n)$ in $\Bbb{Z}$ these functions have the basic property $...
