Accidental Degeneracy of an Elliptic Differential Operator: A Clarification in Terms of Ladder Operators

We consider the linear, second-order elliptic, Schrödinger-type differential operator L:=−12∇2+r22. Because of its rotational invariance, that is it does not change under SO(3) transformations, the eigenvalue problem −12∇2+r22f(x,y,z)=λf(x,y,z) can be studied more conveniently in spherical polar coordinates. It is already known that the eigenfunctions of the problem depend on three parameters. The so-called accidental degeneracy of L occurs when the eigenvalues of the problem depend on one of such parameters only. We exploited ladder operators to reformulate accidental degeneracy, so as to provide a new way to describe degeneracy in elliptic PDE problems.

Symmetry group of the equiangular cubed sphere

The equiangular cubed sphere is a spherical grid, widely used in computational physics. This paper deals with mathematical properties of this grid. We identify the symmetry group, i.e. the group of the orthogonal transformations that leave the cubed sphere invariant. The main result is that it coincides with the symmetry group of a cube. The proposed proof emphasizes metric properties of the cubed sphere. We study the geodesic distance on the grid, which reveals that the shortest geodesic arcs match with the vertices of a cuboctahedron. The results of this paper lay the foundation for future numerical schemes, based on rotational invariance of the cubed sphere.

Noise Robust Projection Rule for Klein Hopfield Neural Networks

Multistate Hopfield models, such as complex-valued Hopfield neural networks (CHNNs), have been used as multistate neural associative memories. Quaternion-valued Hopfield neural networks (QHNNs) reduce the number of weight parameters of CHNNs. The CHNNs and QHNNs have weak noise tolerance by the inherent property of rotational invariance. Klein Hopfield neural networks (KHNNs) improve the noise tolerance by resolving rotational invariance. However, the KHNNs have another disadvantage of self-feedback, a major factor of deterioration in noise tolerance. In this work, the stability conditions of KHNNs are extended. Moreover, the projection rule for KHNNs is modified using the extended conditions. The proposed projection rule improves the noise tolerance by a reduction in self-feedback. Computer simulations support that the proposed projection rule improves the noise tolerance of KHNNs.

Direction-of-Arrival Estimation for 2D Coherently Distributed Sources with Nested Array Based on Matrix Reconstruction

This paper has made proposition of a nested array and an estimation algorithm for direction-of-arrival (DOA) of two-dimensional (2D) coherently distributed (CD) sources. According to the difference coarray concept, double parallel hole-free virtual uniform linear arrays are generated by virtue of vectorization operation on cross-correlation matrices of subarrays. Sensor coordinates of virtual arrays are derived. Rational invariance relationships of virtual arrays are derived. According to the rotational invariance relationships, matrices satisfying rotation invariance are constructed by extracting and regrouping the receive vectors of the virtual arrays, and then an estimation of signal parameters via rotational invariance techniques- (ESPRIT-) like framework on matrix reconstruction is deduced. Optimal configuration of the nested array as well as computational complexity are analyzed. Without pair matching, the proposed method can resolve more sources than the sensor number. Simulation outcomes indicate that the proposed method tends to have a better performance as compared to the traditional uniform arrays that have similar number of sensors.