In this paper, we introduce the definition of the generalized reduced biquaternions and propose a real representation of a generalized reduced biquaternion matrix. By using the real matrix representation, we discuss the least-squares problems of the classic generalized reduced biquaternion matrix equation AXC = B. The least-squares solution to the above matrix equation is formulated by a least-squares real solution of its corresponding real matrix equation. Furthermore, two numerical examples are given to illustrate our results.

In this paper, we investigate the minimum-norm least squares solution to a quaternion tensor system ${\mathcal{A}}_1{*}_N{\mathcal{X}}_1={\mathcal{C}}_1,$${\mathcal{A}}_1{*}_N{\mathcal{X}}_2+{\mathcal{A}}_2{*}_N{\mathcal{X}}_3={\mathcal{C}}_2,$${\mathcal{E}}_1{*}_N{\mathcal{X}}_1{*}_M{\mathcal{F}}_1+{\mathcal{E}}_1{*}_N{\mathcal{X}}_2{*}_M{\mathcal{F}}_2+{\mathcal{E}}_2{*}_N{\mathcal{X}}_3{*}_M{\mathcal{F}}_2=\mathcal{D}$ by using the Moore–Penrose inverses of block tensors. As an application, we discuss the quaternion tensor system $\mathcal{A}{*}_N\mathcal{X}=\mathcal{C},$$\mathcal{E}{*}_N\mathcal{X}{*}_M\mathcal{F}=\mathcal{D}$ for minimum-norm least squares reducible solutions. To illustrate the results, we present an algorithm and a numerical example.

A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary integral quadratic forms over $k$ . A number field $k$ admits a ternary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is even. In this case, there are infinitely many classes of such ternary integral quadratic forms over $k$ . An integral quadratic form over a number field $k$ with more than one variables represents all integers of $k$ over the ring of integers of a finite extension of $k$ if and only if this quadratic form represents $1$ over the ring of integers of a finite extension of $k$ .

Vector quantized variational autoencoders, as variants of variational autoencoders, effectively capture discrete representations by quantizing continuous latent spaces and are widely used in generative tasks. However, these models still face limitations in handling complex image reconstruction, particularly in preserving high-quality details. Moreover, quaternion neural networks have shown unique advantages in handling multi-dimensional data, indicating that integrating quaternion approaches could potentially improve the performance of these autoencoders. To this end, we propose QVQ-VAE, a lightweight network in the quaternion domain that introduces a quaternion-based quantization layer and training strategy to improve reconstruction precision. By fully leveraging quaternion operations, QVQ-VAE reduces the number of model parameters, thereby lowering computational resource demands. Extensive evaluations on face and general object reconstruction tasks show that QVQ-VAE consistently outperforms existing methods while using significantly fewer parameters.