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edit Topics in Nonlinear Dynamics and Chaos Theory x Text Here

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Test

Type Theory	Logic	Set Theory	Homotopy Theory
$A$	proposition	set	space
$a : A$	proof	element	point
$B(x)$	predicate	family of sets	fibration
$b(x) : B(x)$	conditional proof	family of elements	section
$0, 1$	$\top, \bot$	$\varnothing, \{\varnothing\}$	$\varnothing, \star$
$A + B$	$A \vee B$	disjoint union	coproduct of spaces
$A \times B$	$A \wedge B$	set of pairs	product of spaces
$A \to B$	$A \implies B$	set of functions	function space
$\sum_{(x:A)} B(x)$	$\exists_{x : A}B(x)$	disjoint sum	total space
$\prod_{(x:A)} B(x)$	$\forall_{x : A}B(x)$	product	space of sections
$\mathrm{Id}_x$	equality	$\{(x,x)\; :\; x \in A\}$	path space $\mathcal{P}(A)$

Comparission of Classical and Modern Models

	Classical Model	Deep Learning
Parameters	Identifiable	Nonidentifiable
Algebra	Linear Algebra	Rings and Ideals
Geometry	Manifold	Variety
Asymptotics	Laplace Approximation	Gelfand Zeta Function
Probability	Central limit theorem	Empirical process (Mixtures)
Statistics	Statistical Learning Theory	Singular Learning Theory