/sci/ - Science & Math

>>12563737
>can just simplify everything you said
>as they appear in reality
>why the KP hierarchy should be a physical description of nonlinear flow of shallow fluids
...

>>12522180
Look up Kac-Moody affine Lie alg's and affine Manin triples.
>>12522296
Guest's book has a nice take from the quantum cohomo perspective.
>>12522720
Katz.
>>12523028
A (trivializable) normal bundle determines your orientation, hence which normal direction to take is up to you. However once you've picked one for a segment of your mesh, you can ensure orientability by making sure it does not change sign as you move across the segments.

>>12511039
Oh, ok. Whatever you say.

>>12464676
Yes it does, for the delooping/classifying space [math]BG[/math]. In general for any [math]n[/math]-groupoid [math]G[/math] you have [math][X,BG]\cong H^1(X,G)[/math].
>>12466297
It [math]is[/math] harder.
>>12468857
Four.

>>11549195
On domains with high symmetry you definitely can, as solutions are characterized by their [math]L^2[/math]-reps and you can count the order of the symmetry of the domain by counting degeneracies of each eigenmode. The short answer in general, however, is no. What we know in 2D is that we can hear the shapes of the drums up to a certain "triangulated equivalence": domains are Laplace-isospectral iff solutions local on each 1-simplex can be patched across them satisfying certain consistency and boundary conditions. See https://arxiv.org/abs/1101.1239..
Now intuitively this tells me that the answer would also be no in arbitrary dimension (indeed it is also no in 3D): just topologically, the triangulated domains are characterized by Pachner moves which grow in dimension, and these must be necessarily weaker than the above triangulated equivalence. Hence the space of triangulated equivalent domains would also grow in dimension. This means that we'd be less and less accurate in determining the shape of the drumhead given its sound.

>>11530841
It's a very elegant result of Chern-Weil theory that allows you to write the Chern characteristic class [math]c_q: K^\ast(M) \rightarrow H^\ast(M,\mathbb{Q})[/math] (paired with the fundamental class [math][M]\in H_n(M,\mathbb{Q})[/math]) as a local integral.
Good luck anon.