/sci/ - Science & Math

>>11477631
If a|b and b|a then a = b.

>>11431749
2-torus? I mean it's pretty obvious

>>10016567
Has his proof of the non-existence of a complex structure on the 6-sphere even been confirmed yet?
Wtf are you doing Atiyah

>>9313072
No. Let the set of [math]\Psi:X \rightarrow \mathbb{C}[/math]'s form a Banach space [math]B(X)[/math] of functions on a compact set [math]X[/math], then the set of linear operators [math]\mathcal{B}(B(X)) = \operatorname{End}B(X)[/math] is an operator [math]\mathbb{C}[/math]-algebra which makes [math]B(X)[/math] into a left-[math]\mathbb{C}[/math] module. Given a tensor product structure on the Banach space [math]B(X)
= \mathcal{X}(X)\otimes \mathcal{Y}(X)[/math] the morphisms in [math]\operatorname{End}B(X) = \operatorname{Hom}(\mathcal{X}(X)\otimes \mathcal{Y}(X),B(X)) [/math] factors through this tensor product, meaning that the operator [math]\Omega \in \operatorname{End}B(X)[/math] factors into distinct operators [math]\Omega_{\mathcal{X}}\in \operatorname{Hom}(\mathcal{X}(X),B(X))[/math] and [math]\Omega_{\mathcal{Y}}\in \operatorname{Hom}(\mathcal{Y}(X),B(X))[/math] such that [math]\Omega = \Omega_{\mathcal{X}}\otimes\Omega_{\mathcal{Y}} [/math], but [math]\Omega_{\mathcal{X}}[/math] nor [math]\Omega_{\mathcal{Y}}[/math] will be equal to [math]\Omega[/math] unless [math]\Omega[/math] is the identity. In fact this can be proven from the universal property of tensor structures on monoidal categories.

>>9204195
Think of the path you're integrating through as a path [math]\gamma: [0,1]\rightarrow \mathbb{R}^6[/math] where [math](\mathbb{R}^6,\omega)[/math] is the classical phase space. Time just parameterizes the path, so the total derivative is [math]\frac{dv_i}{dx} = \frac{\partial v_i}{\partial x_k} \dot{x}_k [/math] for coordinate [math]x = x(t)[/math] parameterized by the path [math]\gamma[/math].

>>9026762
>How do we know if the content being found inside mathematics is applicable or if it even works???
How do we know if it does or doesn't if we don't try?!?!?!?!?!