Abstract L. Moret-Bailly constructed families $${\mathfrak {C}}\rightarrow {\mathbb {P}}^1$$ C → P 1 of genus 2 curves with supersingular Jacobian. In this paper we first classify the reducible fibers a family usi...

In this paper, we propose three real representations of a generalized Segre quaternion matrix. We establish necessary and sufficient conditions for the existence ?-anti-Hermitian solution to system constrained matrix equations over algebra. also obtain expression general when it is solvable. Finally, provide numerical example verify main results paper.

A Riemannian manifold is said to be Kähler if the holonomy group is contained in U(n). It is quaternion Kähler if the holonomy group is contained in Sp(n)Sp(1). It is known that quaternion Kähler manifolds of dimension ≥ 8 are Einstein, so the scalar curvature s splits these manifolds according to whether s > 0, s = 0 or s < 0. Ricci flat quaternion Kähler manifolds include hyperkähler manifold...

This paper shows the analysis and design of feedforward neural networks using the coordinate-free system of Clifford or geometric algebra. It is shown that real-, complex-, and quaternion-valued neural networks are simply particular cases of the geometric algebra multidimensional neural networks and that some of them can also be generated using support multivector machines (SMVMs). Particularly...

Edge detection is widely used for image processing to improve the and classification of objects, segmentation, extraction other features. Satellite images are rich in information about objects with different color intensity have a large amount noise, so it difficult achieve recognition, classification, feature small through traditional edge algorithms. The colors satellite suffer from overlap d...

The field of neural networks has seen significant advances in recent years with the development deep and convolutional networks. Although many current works address real-valued models, studies reveal that hypercomplex-valued parameters can better capture, generalize, represent complexity multidimensional data. This paper explores quaternion-valued network application for a pattern recognition t...

We will derive both quaternion and octonion algebras as the Clebsch-Gordan algebras based upon the su(2) Lie algebra by considering angular momentum spaces of spin one and three. If we consider both spin 1 and 1 2 states, then the same method will lead to the Lie-super algebra osp(1,2). Also, the quantum generalization of the method is discussed on the basis of the quantum group suq(2). * Paper...

Up until now we have learned that a rotation in R3 about an axis through the origin can be represented by a 3×3 orthogonal matrix with determinant 1. However, the matrix representation seems redundant because only four of its nine elements are independent. Also the geometric interpretation of such a matrix is not clear until we carry out several steps of calculation to extract the rotation axis...

This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our results with an explicit example. 1. Abelian surfaces with quaternionic multiplication Fix Q an ...