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  1. Hyperbolic neural networks have been popular in the re- cent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in develop- ing these networks lies in the nonlinearity of the embed- ding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group which is a semi-Riemannian manifold, i.e. a mani- fold equipped with an indefinite metric. Most existing meth- ods (with some exceptions) use local linearization to de- fine a variety of operations paralleling those used in tra- ditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network which uses the concept of projections (embeddings) fol- lowed by an intrinsic aggregation and a nonlinearity all within the hyperbolic space. The novelty here lies in the projection which is designed to project data on to a lower- dimensional embedded hyperbolic space and hence leads to a nested hyperbolic space representation independently useful for dimensionality reduction. The main theoretical contribution is that the proposed embedding is proved to be isometric and equivariant under the Lorentz transforma- tions, which are the natural isometric transformations in hyperbolic spaces. This projection is computationally effi- cient since it can be expressed by simple linear operations, and, due to the aforementioned equivariance property, it al- lows for weight sharing. The nested hyperbolic space rep- resentation is the core component of our network and there- fore, we first compare this representation – independent of the network – with other dimensionality reduction methods such as tangent PCA, principal geodesic analysis (PGA) and HoroPCA. Based on this equivariant embedding, we develop a novel fully hyperbolic graph convolutional neural network architecture to learn the parameters of the projec- tion. Finally, we present experiments demonstrating com- parative performance of our network on several publicly available data sets. 
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  3. Abstract

    Solution‐processable highly conductive polymers are of great interest in emerging electronic applications. For p‐doped polymers, conductivities as high a nearly 105S cm−1have been reported. In the case of n‐doped polymers, they often fall well short of the high values noted above, which might be achievable, if much higher charge‐carrier mobilities determined could be realized in combination with high charge‐carrier densities. This is in part due to inefficient doping and dopant ions disturbing the ordering of polymers, limiting efficient charge transport and ultimately the achievable conductivities. Here, n‐doped polymers that achieve a high conductivity of more than 90 S cm−1by a simple solution‐based co‐deposition method are reported. Two conjugated polymers with rigid planar backbones, but with disordered crystalline structures, exhibit surprising structural tolerance to, and excellent miscibility with, commonly used n‐dopants. These properties allow both high concentrations and high mobility of the charge carriers to be realized simultaneously in n‐doped polymers, resulting in excellent electrical conductivity and thermoelectric performance.

     
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