On a remarkable geometric-mechanical synergism based on a novel linear eigenvalue problem

The vertices of two specific eigenvectors, obtained from a novel linear eigenvalue problem, describe two curves on the surface of an N-dimensional unit hypersphere. N denotes the number of degrees of freedom in the framework of structural analysis by the Finite Element Method. The radii of curvature of these two curves are 0 and 1. They correlate with pure stretching and pure bending, respectively, of structures. The two coefficient matrices of the eigenvalue problem are the tangent stiffness matrix at the load level considered and the one at the onset of loading. The goals of this paper are to report on the numerical verification of the aforesaid geometric-mechanical synergism and to summarize current attempts of its extension to combinations of stretching and bending of structures.

Analysis of Gegenbauer kernel filtration on the hypersphere

In this study, we aim to construct explicit forms of convolution formulae for Gegenbauer kernel filtration on the surface of the unit hypersphere. Using the properties of Gegenbauer polynomials, we reformulated Gegenbauer filtration as the limit of a sequence of finite linear combinations of hyperspherical Legendre harmonics and gave proof for the completeness of the associated series. We also proved the existence of a fundamental solution of the spherical Laplace-Beltrami operator on the hypersphere using the filtration kernel. An application of the filtration on a one-dimensional Cauchy wave problem was also demonstrated.

Nearly uniform sampling of crystal orientations

A method is presented for generating nearly uniform distributions of three-dimensional orientations in the presence of symmetry. The method is based on the Thomson problem, which consists in finding the configuration of minimal energy of N electrons located on a unit sphere – a configuration of high spatial uniformity. Orientations are represented as unit quaternions, which lie on a unit hypersphere in four-dimensional space. Expressions of the electrostatic potential energy and Coulomb's forces are derived by working in the tangent space of orientation space. Using the forces, orientations are evolved in a conventional gradient-descent optimization until equilibrium. The method is highly versatile as it can generate uniform distributions for any number of orientations and any symmetry, and even allows one to prescribe some orientations. For large numbers of orientations, the forces can be computed using only the close neighbourhoods of orientations. Even uniform distributions of as many as 106 orientations, such as those required for dictionary-based indexing of diffraction patterns, can be generated in reasonable computation times. The presented algorithms are implemented and distributed in the free (open-source) software package Neper.

Reservoir computing on the hypersphere

Reservoir Computing (RC) refers to a Recurrent Neural Network (RNNs) framework, frequently used for sequence learning and time series prediction. The RC system consists of a random fixed-weight RNN (the input-hidden reservoir layer) and a classifier (the hidden-output readout layer). Here, we focus on the sequence learning problem, and we explore a different approach to RC. More specifically, we remove the nonlinear neural activation function, and we consider an orthogonal reservoir acting on normalized states on the unit hypersphere. Surprisingly, our numerical results show that the system’s memory capacity exceeds the dimensionality of the reservoir, which is the upper bound for the typical RC approach based on Echo State Networks (ESNs). We also show how the proposed system can be applied to symmetric cryptography problems, and we include a numerical implementation.

A Maximizing Model of Spherical Bezdek-Type Fuzzy Multi-Medoids Clustering

This paper proposes three modifications for the maximizing model of spherical Bezdek-type fuzzyc-means clustering (msbFCM). First, we use multi-medoids instead of centroids (msbFMMdd), which is similar to modifying fuzzyc-means to fuzzy multi-medoids. Second, we kernelize msbFMMdd (K-msbFMMdd). msbFMMdd can only be applied to objects in the first quadrant of the unit hypersphere, whereas its kernelized form can be applied to a wider class of objects. The third modification is a spectral clustering approach to K-msbFMMdd using a certain assumption. This approach improves the local convergence problem in the original algorithm. Numerical examples demonstrate that the proposed methods can produce good results for clusters with nonlinear borders when an adequate parameter value is selected.