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The ultrapower $T^{\ast}$ of an arbitrary ordered set $T$ is introduced as an infinitesimal extension of $T$. It is obtained as the set of equivalence classes of the sequences in $T$, where the corresponding relation is generated by an ultrafilter on the set of natural numbers. It is established that $T^{\ast}$ always satisfies Cantor's property, while one can give the necessary and sufficient conditions for $T$ so that $T^{\ast}$ would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of $T^{\ast}$ is determined by the cardinality of $T$., Comment: 10 pages

This paper presents a new family of backpropagation-free neural architectures, Gated Linear Networks (GLNs). What distinguishes GLNs from contemporary neural networks is the distributed and local nature of their credit assignment mechanism; each neuron directly predicts the target, forgoing the ability to learn feature representations in favor of rapid online learning. Individual neurons can model nonlinear functions via the use of data-dependent gating in conjunction with online convex optimization. We show that this architecture gives rise to universal learning capabilities in the limit, with effective model capacity increasing as a function of network size in a manner comparable with deep ReLU networks. Furthermore, we demonstrate that the GLN learning mechanism possesses extraordinary resilience to catastrophic forgetting, performing comparably to a MLP with dropout and Elastic Weight Consolidation on standard benchmarks. These desirable theoretical and empirical properties position GLNs as a complementary technique to contemporary offline deep learning methods., Comment: arXiv admin note: substantial text overlap with arXiv:1712.01897

This paper introduces equivariant hamiltonian flows, a method for learning expressive densities that are invariant with respect to a known Lie-algebra of local symmetry transformations while providing an equivariant representation of the data. We provide proof of principle demonstrations of how such flows can be learnt, as well as how the addition of symmetry invariance constraints can improve data efficiency and generalisation. Finally, we make connections to disentangled representation learning and show how this work relates to a recently proposed definition.

The Hamiltonian formalism plays a central role in classical and quantum physics. Hamiltonians are the main tool for modelling the continuous time evolution of systems with conserved quantities, and they come equipped with many useful properties, like time reversibility and smooth interpolation in time. These properties are important for many machine learning problems - from sequence prediction to reinforcement learning and density modelling - but are not typically provided out of the box by standard tools such as recurrent neural networks. In this paper, we introduce the Hamiltonian Generative Network (HGN), the first approach capable of consistently learning Hamiltonian dynamics from high-dimensional observations (such as images) without restrictive domain assumptions. Once trained, we can use HGN to sample new trajectories, perform rollouts both forward and backward in time and even speed up or slow down the learned dynamics. We demonstrate how a simple modification of the network architecture turns HGN into a powerful normalising flow model, called Neural Hamiltonian Flow (NHF), that uses Hamiltonian dynamics to model expressive densities. We hope that our work serves as a first practical demonstration of the value that the Hamiltonian formalism can bring to deep learning.

Single- and few-layered InSe flakes are produced by the liquid-phase exfoliation of beta-InSe single crystals in 2-propanol, obtaining stable dispersions with a concentration as high as 0.11 g/L. Ultracentrifugation is used to tune the morphology, i.e., the lateral size and thickness of the as-produced InSe flakes. We demonstrate that the obtained InSe flakes have maximum lateral sizes ranging from 30 nm to a few um, and thicknesses ranging from 1 to 20 nm, with a max population centred at ~ 5 nm, corresponding to 4 Se-In-In-Se quaternary layers. We also show that no formation of further InSe-based compounds (such as In2Se3) or oxides occurs during the exfoliation process. The potential of these exfoliated-InSe few-layer flakes as a catalyst for hydrogen evolution reaction (HER) is tested in hybrid single-walled carbon nanotubes/InSe heterostructures. We highlight the dependence of the InSe flakes morphologies, i.e., surface area and thickness, on the HER performances achieving best efficiencies with small flakes offering predominant edge effects. Our theoretical model unveils the origin of the catalytic efficiency of InSe flakes, and correlates the catalytic activity to the Se vacancies at the edge of the flakes., Comment: 39 pages

We report on the preparation of polycarbonate-based graphene (PC/G) composites, by using a simple and scalable solution blending method to disperse single (SLG) and few layer (FLG) graphene flakes, prepared by liquid phase exfoliation (LPE), in the polymer matrix. A solvent-exchange process is carried out to re-disperse the exfoliated SLG/FLG flakes in an environmentally friendly solvent, i.e. 1,3dioxolane, which is also used to dissolve the polycarbonate pellets, thus facilitating the mixing of the polymer dispersion with the SLG/FLG flakes. The loading of SLG/FLG flakes improves the mechanical and thermal properties, as well as the electrical conductivity of the polymer, reaching a +26% improvement of the elastic modulus at 1 wt% loading, and an electrical conductivity 10^-3 S m^-1 at 10 wt% with a percolation threshold achieved at 0.55 vol%. The as-prepared PC/G composite with the aforementioned reinforced properties can be a promising material for 3D printing-based applications.

Efficient and scalable production of two-dimensional (2D) materials is required to overcome technological hurdles towards the creation of a 2D-materials-based industry. Here, we present a novel approach developed for the exfoliation of layered crystals, i.e., graphite, hexagonal boron nitride and transition metal dichalcogenides. The process is based on high-pressure wet-jet-milling (WJM), resulting in 2 L/hr production of 10 g/L of single- and few-layer 2D crystal flakes in dispersion making the scaling-up more affordable. The WJM process enables the production of defect-free and high-quality 2D-crystal dispersions on a large scale, opening the way for the full exploitation in different commercial applications, e.g., anodes active material in lithium ion batteries, reinforcement in polymer-graphene composites, and transparent conductors as we demonstrate in this report., Comment: 30 pages, 15 figures

This paper describes a family of probabilistic architectures designed for online learning under the logarithmic loss. Rather than relying on non-linear transfer functions, our method gains representational power by the use of data conditioning. We state under general conditions a learnable capacity theorem that shows this approach can in principle learn any bounded Borel-measurable function on a compact subset of euclidean space; the result is stronger than many universality results for connectionist architectures because we provide both the model and the learning procedure for which convergence is guaranteed., Comment: 40 pages