/sci/ - Science & Math

>>9400465
>can't understand basic quantum mechanics
>wants to study string theory

>>9292288
Let [math]M[/math] be a closed 2-dimensional manifold and [math]U_\alpha[/math] be an open cover with [math]\phi_\alpha:U_\alpha \rightarrow\mathbb{R}^2[/math] homeomorphisms. Let [math]f:\mathbb{R}^2 \rightarrow (0,\infty)[/math] and put [math]f_\alpha = f \circ \phi_\alpha[/math], then by the gluing lemma one can patch together [math]f_\alpha[/math] with the coordinate transition functions to form a continuous function [math]g:M \rightarrow (0,\infty)[/math]. Now put [math]\mathfrak{f} = g|_{[0,1]}[/math], then [math]\mathfrak{f}[/math] is a Morse function and hence its degree is proportional to its Euler chaaracter, and by the Gauss-Bonnet theorem [eqn]\chi(M) \propto \int_{\mathbb{R}^2} d^2x \sum_i \delta(x-x_i),[/eqn] where [math]x_i[/math] are the regular singular points of the pullback [math]\mathfrak{f}^*[/math] along [math]\phi_\alpha[/math]'s onto [math]\mathbb{R}^2[/math]. This means that you can construct, for [math]M[/math] such that [math]\chi(M) = 1[/math], a function [math]g[/math] such that [math]\mathcal{f}[/math] has one local max in the interval [math][0,1][/math], and you can be sure that they cannot have a local min, since that'd make the Euler characteristic of [math]M[/math] less than 1. By picking infinitely many [math]M[/math] with this property, you can construct infinitely many such [math]g_i[/math]'s from [math]M_i\rightarrow [0,1][/math].
The tough part is the issue of gluing these [math]g[/math]'s together such that [math]g_i[/math] maps to [math][i,i+1][/math]. If you can do it then you can just pick patches [math]U_{\alpha,i}[/math] to restrict to and then precompose it with [math]\phi_{\alpha,i}[/math] to obtain the desired map [math]f:\mathbb{R}^2 \rightarrow [0,\infty)[/math].

>>9069545
What is your research? Quantum computing?