/sci/ - Science & Math

>>12001708
>no one "studies the trace" just like no one "studies the chain rule"
What do you mean by that, though? I mean people write books and papers about trace class operators. Or partial traces, in which case they correspond to "entangled" marginalization. So it's definately studied.

The relative change of a parametrized volume [math] \det(A) [/math] being given as

[math] \det \left( A^{-1}\,A'(t) \right) = \operatorname{tr} \left(A(t)^{-1}\,A'(t)\right) [/math]

is maybe a good intuition already.

There's this relation [math] \det(E-A)=\exp(-\sum_n\frac{1}{n}\mathrm{tr}(A^n)) [/math] that pops up occationally,

From

[math] (1-a_p) = \exp\log(1-a_p) = {\mathrm e}^{-\sum_n a_p^n/n} = \prod_n{\mathrm e}^{-a_p^n/n} [/math]

such that

[math] \prod_p^d (1-a_p) = \prod_p^d \prod_n {\mathrm e}^{-a_p^n/n} = \prod_n {\mathrm e}^{-\frac{1}{n}\sum_p^da_p^n} = {\mathrm e}^{-\sum_n\frac{1}{n}\sum_p^da_p^n} [/math]

E.g. as

[math] \zeta(s) = \prod_p^\infty\frac{1}{1-p^{-s}} \implies \zeta(s) = \exp\left(\sum_n\frac{1}{n}\sum_p^\infty p^{-s\cdot{}n}\right) [/math]

or local zeta-function's. It made me think the rationality part of the Weil conjectures aren't as far fetched as I first thought.

>nonfirstorderizability

>plurisubharmonic

>Führerdiskriminantenproduktformel

>deterritorialization

>Rindfleischetikettierungsüberwachungsaufgabenübertragungsgesetz

...Okay the last two aren't STEM

nonfirstorderizability
plurisubharmonic
Führerdiskriminantenproduktformel
deterritorialization
Rindfleischetikettierungsüberwachungsaufgabenübertragungsgesetz

Okay the last two aren't STEM

sorry for reposting this, if you replied to it in the meantime. I don't want to have people more confused with half-finished sentences

>>11629716
>>11629730
I can't tell who's who but best not use my P(n) as a set, I don't know why
>>11629716
says
> bijection f between \omega_1 and P(N)
or why
>>11629730
says P isn't in the language of set theory.
Both of your claims seem wrong in that instances

>>11629704
Well I'm not going to expands the definition of \omega_1 for you guys.

But let me see how far I can get with your or his (who ever of you is the "vacuuously true guy" Shyamalan rhetoric guys) claim.
I suppose it's >>11629720

Say w, r are two set variables and Ind(w) and U(r) say that w is the smallest inductive set and r is the first uncountable sets.

The idea, I understand, is to recast CH as

forall r, w. U(r) => [Ind(w) => ch(w, r)]

where little "ch" means
|w -> {0,1}| = |r|
where "=" is the existence claim of a bijection

So dissolving the implications under the forall as
P=>Q ... ]not P] or Q,
then CH is

forall r, w. [not U(r)] or [[not Ind(w)] or ch(w, r)]

i.e.

forall r, w. [not U(r)] or [not Ind(w)] or ch(w, r)]

and if we take the second disjunct as an axiom, we should be done.

done with simplifying anyway, tell me if that kind of formulation is what you wanted to get at or where the misinterpretation is
?

>>11629716
>>11629730
I can't tell who's who but best not use my P(n) as a set, I don't know why
>>11629716
says
> bijection f between \omega_1 and P(N)
or why
>>11629730
says P isn't in the language of set theory.
Both of your claims seem wrong in that instances

>>11629704
Well I'm not going to expands the definition of \omega_1 for you guys.

But let me see how far I can get with your or his (who ever of you is the "vacuuously true guy" Shyamalan rhetoric guys) claim.
I suppose it's >>11629720

Say w, r are two set variables and Ind(w) and U(r) say that w is the smallest inductive set and r is the first uncountable sets.

The idea, I understand, is to recast CH as

forall r, w. U(r) => [Ind(w) => ch(w, r)]

where little "ch" means
|w -> {0,1}| = |r|
where "=" is the existence claim of a bijection

So dissolving the implications under the forall as
P=>Q ... ]not P] or Q,
then CH is

forall r, w. [not U(r)] or [[not Ind(w)] or ch(w, r)]

i.e.

forall r, w. [not U(r)] or [not Ind(w)] or ch(w, r)]

and if we take the second disjunct as an axiom, we should be done.