Polymers with spatial or topological constraints: Theoretical and computational results

In this review, we provide an organized summary of the theoretical and computational results that are available for polymers subject to spatial or topological constraints. Because of the interdisciplinary character of the topic, we provide an accessible, non-specialist introduction to the main topological concepts, polymer models, and theoretical/computational methods used to investigate dense and entangled polymer systems. The main body of our review deals with (i) the effect that spatial confinement has on the equilibrium topological entanglement of one or more polymer chains and (ii) the metric and entropic properties of polymer chains with fixed topological states. These problems have important technological applications and implications for life sciences. Both aspects, especially the latter, are amply covered. A number of selected open problems are finally highlighted.     

Approximation in weighted p-mean by RBF networks of Delsarte translates

We give conditions on the kernel function (or activation function) for the family of radial basis function (RBF) neural networks obtained upon replacing the usual translation by the Delsarte one, with not necessarily the same smoothing factor in all kernel nodes, to have the universal approximation property in suitable weighted L^p$L^p$-spaces (1?p<∞)$(1?p<\infty )$. A complete characterization of such kernels for p=1$p=1$ and p=2$p=2$ is provided.     

Lifting via cocycle deformation

We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such lifting.     

Crystal energy functions via the charge in types A and C

The Ram–Yip formula for Macdonald polynomials (at t = 0) provides a statistic which we call charge. In types A and C it can be defined on tensor products of Kashiwara–Nakashima single column crystals. In this paper we prove that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial R-matrix.