Abstract:
We show that, in quaternion quantum mechanics with a complex geometry, the minimal four Higgs of the unbroken electroweak theory naturally determine the quaternion invariance group which corresponds to the Glashow group. Consequently, we are able to identify the physical significance of the anomalous Higgs scalar solutions. We introduce and discuss the complex projection of the Lagrangian density.

Abstract:
In recent years, applications of quaternion matrices are getting more and more important and extensive in quantum mechanics and rigid mechanics. With the rapid development of the above disciplines, it becomes necessary for us to further study the theories of quaternion matrices, but the studies and applications are more difficult due to the non-commutation of quaternion,. In this paper, a new mathematical algebra method of quaternionic mechanics was presented. In the study of mathematical methods of quaternionic mechanics, a concept of companion vector was introduced and some properties of the companion vector were discussed. By the complex presentation of quaternion, this paper introduced new concepts of determinant and rank of quaternion matrices, studied some properties of determinant, rank, invertible matrix, characteristic value and characteristic vector and linear equations over quaternion field, and derived some simple algebraic methods of above subjects, and obtained simple Cramer's rule and Cayley-Hamilton theorem over quaternion field.

Abstract:
We complete the rules of translation between standard complex quantum mechanics (CQM) and quaternionic quantum mechanics (QQM) with a complex geometry. In particular we describe how to reduce ($2n$+$1$)-dimensional complex matrices to {\em overlapping\/} ($n$+$1$)-dimensional quaternionic matrices with generalized quaternionic elements. This step resolves an outstanding difficulty with reduction of purely complex matrix groups within quaternionic QM and avoids {\em anomalous} eigenstates. As a result we present a more complete translation from CQM to QQM and viceversa.

Abstract:
The goal of this study is to present quaternion Kaehler analogue of Hamiltonian mechanics. Finally, the some results related to quaternion Kaehler dynamical systems were also given.

Abstract:
The aim of this study is to introduce quaterinon Kaehler analogue of Lagrangian mechanics. Finally, the geometric and physical results related to quaternion Kaehler dynamical systems are also presented.

Abstract:
The Lee-Friedrichs model has been very useful in the study of decay-scattering systems in the framework of complex quantum mechanics. Since it is exactly soluble, the analytic structure of the amplitudes can be explicitly studied. It is shown in this paper that a similar model, which is also exactly soluble, can be constructed in quaternionic quantum mechanics. The problem of the decay of an unstable system is treated here. The use of the Laplace transform, involving quaternion-valued analytic functions of a variable with values in a complex subalgebra of the quaternion algebra, makes the analytic properties of the solution apparent; some analysis is given of the dominating structure in the analytic continuation to the lower half plane. A study of the corresponding scattering system will be given in a succeeding paper.

Abstract:
We present the two-dimensional quaternion Wigner-Ville distribution (QWVD). The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the classical Wigner-Ville distribution definition. Based on the properties of quaternions and the QFT kernel we obtain three types of the QWVD. We discuss some useful properties of various definitions for the QWVD, which are extensions of the classical Wigner-Ville distribution properties. 1. Introduction The classical Wigner-Ville distribution (WVD) or Wigner-Ville transform (WVT) is an important tool in the time-frequency signal analysis. It was first introduced by Eugene Wigner in his calculation of the quantum corrections of classical statistical mechanics. It was independently derived again by J. Ville in 1948 as a quadratic representation of the local time-frequency energy of a signal. In [1–3], the authors introduced the WVT and established some important properties of the WVT. The transform is then extended to the linear canonical transform (LCT) domain by replacing the kernel of the classical Fourier transform (FT) with the kernel of the LCT in the WVD domain [4]. As a generalization of the real and complex Fourier transform (FT), the quaternion Fourier transform (QFT) has been of interest to researchers for some years. A number of useful properties of the QFT have been found including shift, modulation, convolution, correlation, differentiation, energy conservation, uncertainty principle, and so on. Due to the noncommutative property of quaternion multiplication, there are three different types of two-dimensional QFTs. These three QFTs are so-called a left-sided QFT, a right-sided QFT, and a two-sided QFT, respectively (see, e.g., [5–9]). In [10, 11], special properties of the asymptotic behaviour of the right-sided QFT are discussed and generalization of the classical Bohner-Millos theorems to the framework of quaternion analysis is established. Many generalized transforms are closely related to the QFTs, for example, the quaternion wavelet transform, fractional quaternion Fourier transform, quaternion linear canonical transform, and quaternionic windowed Fourier transform [12–18]. Based on the QFTs, one also may extend the WVD to the quaternion algebra while enjoying similar properties as in the classical case. Therefore, the main purpose of this paper is to propose a generalization of the classical WVD to quaternion algebra, which we call the quaternion Wigner-Ville distribution (QWVD). Our generalization is constructed by substituting the

Abstract:
The polynomials with quaternion coefficients have two kind of roots: isolated and spherical. A spherical root generates a class of roots which contains only one complex number $z$ and its conjugate $\bar{z}$, and this class can be determined by $z$. In this paper, we deal with the complex roots of quaternion polynomials. More precisely, using B\'{e}zout matrices, we give necessary and sufficient conditions, for a quaternion polynomial to have a complex root, a spherical root, and a complex isolated root. Moreover, we compute a bound for the size of the roots of a quaternion polynomial.

Abstract:
The paper aims to adopt the complex quaternion and octonion to formulate the field equations for electromagnetic and gravitational fields. Applying the octonionic representation enables one single definition to combine some physics contents of two fields, which were considered to be independent of each other in the past. J. C. Maxwell applied simultaneously the vector terminology and the quaternion analysis to depict the electromagnetic theory. This method edified the paper to introduce the quaternion and octonion spaces into the field theory, in order to describe the physical feature of electromagnetic and gravitational fields, while their coordinates are able to be the complex number. The octonion space can be separated into two subspaces, the quaternion space and the S-quaternion space. In the quaternion space, it is able to infer the field potential, field strength, field source, field equations, and so forth, in the gravitational field. In the S-quaternion space, it is able to deduce the field potential, field strength, field source, and so forth, in the electromagnetic field. The results reveal that the quaternion space is appropriate to describe the gravitational features; meanwhile the S-quaternion space is proper to depict the electromagnetic features.

Abstract:
This article investigates Kak neural networks, which can be instantaneously trained, for complex and quaternion inputs. The performance of the basic algorithm has been analyzed and shown how it provides a plausible model of human perception and understanding of images. The motivation for studying quaternion inputs is their use in representing spatial rotations that find applications in computer graphics, robotics, global navigation, computer vision and the spatial orientation of instruments. The problem of efficient mapping of data in quaternion neural networks is examined. Some problems that need to be addressed before quaternion neural networks find applications are identified.