Spherical functions and local densities on the space of p-adic quaternion Hermitian forms

We introduce the space [Formula: see text] of quaternion Hermitian forms of size [Formula: see text] on a [Formula: see text]-adic field with odd residual characteristic, and define typical spherical functions [Formula: see text] on [Formula: see text] and give their induction formula on sizes by using local densities of quaternion Hermitian forms. Then, we give functional equation of spherical functions with respect to [Formula: see text], and define a spherical Fourier transform on the Schwartz space [Formula: see text] which is Hecke algebra [Formula: see text]-injective map into the symmetric Laurent polynomial ring of size [Formula: see text]. Then, we determine the explicit formulas of [Formula: see text] by a method of the author’s former result. In the last section, we give precise generators of [Formula: see text] and determine all the spherical functions for [Formula: see text], and give the Plancherel formula for [Formula: see text].

SOBOLEV SPACES ARISING FROM A GENERALIZED SPHERICAL FOURIER TRANSFORM

In this paper, we define Sobolev spaces on a locally compact unimodular group in link with the spherical Fourier transform of type $\delta$. Properties of these spaces are obtained. Analogues of Sobolev embedding theorems are proved.

Machine-learned interatomic potentials for alloys and alloy phase diagrams

We introduce machine-learned potentials for Ag-Pd to describe the energy of alloy configurations over a wide range of compositions. We compare two different approaches. Moment tensor potentials (MTPs) are polynomial-like functions of interatomic distances and angles. The Gaussian approximation potential (GAP) framework uses kernel regression, and we use the smooth overlap of atomic position (SOAP) representation of atomic neighborhoods that consist of a complete set of rotational and permutational invariants provided by the power spectrum of the spherical Fourier transform of the neighbor density. Both types of potentials give excellent accuracy for a wide range of compositions, competitive with the accuracy of cluster expansion, a benchmark for this system. While both models are able to describe small deformations away from the lattice positions, SOAP-GAP excels at transferability as shown by sensible transformation paths between configurations, and MTP allows, due to its lower computational cost, the calculation of compositional phase diagrams. Given the fact that both methods perform nearly as well as cluster expansion but yield off-lattice models, we expect them to open new avenues in computational materials modeling for alloys.

A novel scan registration method based on the feature-less global descriptor – spherical entropy image

Purpose This paper aims to present the spherical entropy image (SEI), a novel global descriptor for the scan registration of three-dimensional (3D) point clouds. This paper also introduces a global feature-less scan registration strategy based on SEI. It is advantageous for 3D data processing in the scenarios such as mobile robotics and reverse engineering. Design/methodology/approach The descriptor works through representing the scan by a spherical function named SEI, whose properties allow to decompose the six-dimensional transformation into 3D rotation and 3D translation. The 3D rotation is estimated by the generalized convolution theorem based on the spherical Fourier transform of SEI. Then, the translation recovery is determined by phase only matched filtering. Findings No explicit features and planar segments should be contained in the input data of the method. The experimental results illustrate the parameter independence, high reliability and efficiency of the novel algorithm in registration of feature-less scans. Originality/value A novel global descriptor (SEI) for the scan registration of 3D point clouds is presented. It inherits both descriptive power of signature-based methods and robustness of histogram-based methods. A high reliability and efficiency registration method of scans based on SEI is also demonstrated.