Exponential ReLU DNN Expression of Holomorphic Maps in High Dimension

VC dimension of ReLU networks

Tangencies between holomorphic maps and holomorphic laminations

We prove that the set of leaves of a holomorphic lamination of codimension one that are tangent to a germ of a holomorphic map is discrete. Let F be a holomorphic lamination of codimension one in an open set V in a complex Banach space B. In this paper, this means that V = W × C, where W is a neighborhood of the origin in some Banach space, and the leaves Lλ of the lamination are disjoint graph...

Extensions of Holomorphic Maps through Hypersurfaces and Relations to the Hartogs Extensions in Infinite Dimension

A generalization of Kwack’s theorem to the infinite dimensional case is obtained. We consider a holomorphic map f from Z \ H into Y , where H is a hypersurface in a complex Banach manifold Z and Y is a hyperbolic Banach space. Under various assumptions on Z, H and Y we show that f can be extended to a holomorphic map from Z into Y . Moreover, it is proved that an increasing union of pseudoconve...

Normal Holomorphic Maps from C

A theorem of A. Ostrowski describing meromorphic functions f such that the family {f(λz) : λ ∈ C} is normal, is generalized to holomorphic maps from C∗ to a projective space. Let f : C → P be a holomorphic curve and (1) F = (g0, g1, . . . , gn) some homogeneous representation of f . This means that gj are analytic functions in C without common zeros. When n = 1, we have P = C, and f can be iden...

Hausdorff Dimension in Exponential Time

In this paper we investigate effective versions of Hausdorff dimension which have been recently introduced by Lutz. We focus on dimension in the class E of sets computable in linear exponential time. We determine the dimension of various classes related to fundamental structural properties including different types of autoreducibility and immunity. By a new general invariance theorem for resour...