Antiholomorphic function

Abstract: We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function $\sigma : \mathbb{C} \to \mathbb{C}$ in which each neuron performs the operation $\mathbb{C}^N \to \mathbb{C}, z \mapsto \sigma(b + w^T z)$ with weights $w \in \mathbb{C}^N$ and a bias $b \in \mathbb{C}$, and with $\sigma$ applied componentwise. We completely characterize those activation functions $\sigma$ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function on any compact subset of $\mathbb{C}^d$ arbitrarily well. Unlike the classical case of real networks, the set of "good activation functions" which give rise to networks with the universal approximation property differs significantly depending on whether one considers deep networks or shallow networks: For deep networks with at least two hidden layers, the universal approximation property holds as long as $\sigma$ is neither a polynomial, a holomorphic function, or an antiholomorphic function. Shallow networks, on the other hand, are universal if and only if the real part or the imaginary part of $\sigma$ is not a polyharmonic function.

Abstract: It is well-known that the classical concept of modular forms may be introduced as follows Write G for one of the two algebraic groups GL(2), SL(2); take for k the field Q of rational numbers, R being then the completion k ∞ of k at its infinite place; let Γ be the subgroup G Z of G R (ie the group of the matrices in M2 (Z) with the determinant ±1 if G = GL(2), and + 1 if G = SL(2)) On G R , consider the complex-valued functions which are left-invariant under Γ (or at any rate under some congruence subgroup of Γ), behave in a prescribed manner under a translation belonging to the center of G R , and behave in a prescribed manner under the right translations belonging to the usual maximal compact subgroup of G R and under the Casimir operator for G R ; the two latter conditions ensure that this determines in the upper half-plane a modular form of prescribed degree which is an eigenfunction for the Beltrami operator (in particular, if the corresponding eigenvalue is O, it is holomorphic, or at any rate the sum of a holomorphic and an antiholomorphic function)

Abstract: The members of one explicit class of functions in ℂ2 are identified with the geodetic shear-free null congruences in Minkowski's space-time. Members of a second explicit class are identified with the type-N vacuum space-times with twist-free rays. These two classes are special subclasses from a larger class of functions associated with the type-N space-times. This larger class is characterized in the following way: Ifϕ andζ are holomorphic variables in ℂ2, thenu (ϕ, ζ, $$\bar \zeta $$ ), a function holomorphic inϕ, belongs to the class provided the function $$\partial _{\zeta } \partial _{\zeta } $$ ∂ ϕ u/∂ ϕ u satisfies the tangential Cauchy-Riemann equation for an antiholomorphic function on the 3-surface whereu (ϕ, ζ, $$\bar \zeta $$ ) has real values.

Abstract: The conformal invariance properties of the QCD Pomeron in the transverse plane allow us to give an explicit analytical expression for the solution of the BFKL equations both in the transverse coordinate and momentum spaces. This result is obtained from the solution of the conformal eigenvectors in the mixed representation in terms of two conformal blocks, each block being the product of an holomorphic times an antiholomorphic function. This property is used to give an exact expression for the QCD dipole multiplicities and dipole-dipole cross-sections in the whole parameter space, proving the equivalence between the BFKL and dipole representations of the QCD Pomeron.