Any single-hidden-layer feedforward network based on Gaussian or asymptotically constant odd or even rational non-polynomial activation functions has the same property as such networks based on hyperbolic tangent: input-output function determines weights and biases up to a permutation of the hidden units and sign-flips.

For complex parameters a, c, we consider the Hénon mapping Ha,c : C → C, given by (x, y) 7→ (x + c−ay, x), and its Julia set, J . In this paper, we describe a rigorous computer program for attempting to construct a cone field in the tangent bundle over J , which is preserved by DH, and a continuous norm in which DH (and DH) uniformly expands the cones (and their complements). We show a conseque...

This paper studies controllability for the class of control systems commonly called (continuous-time) recurrent neural networks. It is shown that, under a generic condition on the input matrix, the system is controllable, for every possible state matrix. The result holds when the activation function is the hyperbolic tangent.

The Descartes circle theorem states that if four circles are mutually tangent in the plane, with disjoint interiors, then their curvatures (or “bends”) bi = 1 ri satisfy the relation (b1 + b2 + b3 + b4) 2 = 2(b1 + b 2 2 + b 2 3 + b 2 4). We show that similar relations hold involving the centers of the four circles in such a configuration, coordinatized as complex numbers, yielding a complex Des...

We compute the trace formula for magnetic Laplacian on a compact hyperbolic surface of constant curvature with field energies above Mane critical level corresponding geodesic flow. discuss asymptotic behavior coefficients when energy approaches level.

Chaos in a fractional-order jerk model using hyperbolic tangent nonlinearity is presented. A fractional integrator in the model is approximated by linear transfer function approximation in the frequency domain. Resulting chaotic attractors are demonstrated with the system orders as low as 2.1.

We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.

— We consider a partially hyperbolic set K on a Riemannian manifold M whose tangent space splits as TKM = Ecu⊕Es, for which the centre-unstable direction E expands non-uniformly on some local unstable disk. We show that under these assumptions f induces a Gibbs-Markov hyperbolic structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need...

New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harmonic maps from 3-dimensional Euclidean domains to the 2-sphere. Finally, the ...

The 13C NMR chemical shift of sp2 carbon atoms in acyclic alkenes was estimated with multilayer feedforward artificial neural networks (ANNs) and multilinear regression (MLR), using as structural descriptors a vector made of 12 components encoding the environment of the resonating carbon atom. The neural network quantitative model provides better results than the MLR model calibrated with the s...