Search

Due to its applications in signal analysis and image processing, the quaternionic Fourier analysis has received increasing attention. In particular, quaternionic Gabor frames (QGFs) attracted some mathematicians' interest. From the literatures, some results on QGFs are based on quaternionic Fourier orthonormal bases. But those used so-called quaternionic Fourier orthonormal bases have a gap that they are all incomplete. In this paper, we present a class of quaternionic Fourier orthonormal bases, and using them derive the corresponding Gabor orthonormal bases. [ABSTRACT FROM AUTHOR]

The aim of this study is the determination of the best fit ellipsoid to given points by quaternions. The problem of the fitting ellipsoid is frequently encountered in image processing, computer games, medicine, engineering and science applications, geodesy, etc. The ellipsoid fitting problem is the process of determining the ellipsoid that best fits a given set of points in 3D. In the fitting process, it is generally done over two models. The first of these is the algebraic method and the second one is orthogonal (geometric) method. In this study, we tried to solve the problem of algebraic and orthogonal ellipsoid fitting based on Euler angles for the first time over quaternions. The superiority of quaternions over Euler rotation angles is well known. In addition, the variance–covariance matrix of the parameters of the fitted ellipsoid will also be calculated. Numerical applications show that the proposed method can be used successfully. [ABSTRACT FROM AUTHOR]

The asymptotic synchronization of quaternion-valued delayed neural networks with impulses and inertia is studied in this article. Firstly, a convergence result on piecewise differentiable functions is developed, which is a generalization of the Barbalat lemma and provides a powerful tool for the convergence analysis of discontinuous systems. To achieve synchronization, a constant gain-based control scheme and an adaptive gain-based control strategy are directly proposed for response quaternion-valued models. In the convergence analysis, a direct analysis method is developed to discuss the synchronization without using the separation technique or reduced-order transformation. In particular, some Lyapunov functionals, composed of the state variables and their derivatives, are directly constructed and some synchronization criteria represented by matrix inequalities are obtained based on quaternion theory. Some numerical results are shown to further confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this work, we present an accurate and quaternion-based solution for the Perspective-n-Point method for determining the position and orientation of a calibrated camera from a set of n 3D points and their corresponding 2D projections on the image plane referred to as AQPnP. Most of the previous PnP methods have not handled the uncertainty of image feature points directly. In the proposed method, uncertainty of image points is integrated into PnP method. This is achieved by mapping the covariance (uncertainty) matrices of the image points to PnP method. It then uses the alternating direction method of multipliers (ADMM) method to minimize the modeled cost function, which has the ability to find the desired solution in real time. The ADMM method solves the system of equations in a manner that is convergent to the global optimal. The ADMM, a widely used operator splitting technique, has drawn significant attention due to its excellent performance in various applications. Quaternion-based rotation parameterization allows AQPnP to be highly accurate in estimating the rotation matrix. We compared AQPnP method with the state-of-the-art PnP methods, and pose estimation results show that our proposed method outperforms better than other methods. [ABSTRACT FROM AUTHOR]

In light of the significance of non-commutative quaternionic algebra in modern physics, this study proposes the existence of the Klein paradox in the quaternionic (3+1)-dimensional space-time structure. By introducing quaternionic wave function, we rewrite the Klein–Gordon equation in extended quaternionic form that includes scalar and the vector fields. Because quaternionic fields are non-commutative, the quaternionic Klein–Gordon equation provides three separate sets of the probability density and probability current density of relativistic particles. We explore the significance of these probability densities by determining the reflection and transmission coefficients for the quaternionic relativistic step potential. Furthermore, we also discuss the quaternionic version of the oscillatory, tunnelling, and Klein zones for the quaternionic step potential. The Klein paradox occurs only in the Klein zone when the impacted particle's kinetic energy is less than 0 − m 0 c 2 . Therefore, it is emphasized that for the quaternionic Klein paradox, the quaternionic reflection coefficient becomes exclusively higher than value one while the quaternionic transmission coefficient becomes lower than zero. [ABSTRACT FROM AUTHOR]

To be a quantitative and testable tectonic model, plate tectonics requires spherical geometry and spherical kinematics in terms of finite rotations conveniently parametrized by their angle and axis and described by unit quaternions. In treatises on "Plate Tectonics" infinitesimal, instantaneous, and finite rotations, absolute and relative rotations are said to be applied to model the motion of tectonic plates. Even though these terms are strictly defined in mathematics, they are often casually used in geosciences. Here, their definitions are recalled and clarified as well as the terms rotation, orientation, and location on the sphere. For instance, infinitesimal rotations refer to a mathematical limit, when the angle of rotation tends to zero. Their rules do not apply to finite rotations, no matter how small their finite angles of rotation are. Mathematical approaches applying appropriate and feasible assumptions to model spherical motion of tectonic plates over geological times of hundreds of millions of years are derived including (i) sequences of incremental finite rotations, (ii) sequences of accumulating successive concatenations of finite rotations, and (iii) continuous rotations in terms of fully transient quaternions. The incremental and the accumulating approaches provide complementary views. While the relative Euler pole appears to migrate in the latter, it appears fixed in the former. Path, mean, and instantaneous velocity of the migrating Euler pole are derived as well as the angular and trajectoral velocity of the rotational motion about it. The approaches are illustrated by a geological example with actual data and a numerical yet geologically inspired example with artificial data. The former revisits the three‐plate scenario with stationary axes of two "absolute" rotations implying transient "relative" rotations about a migrating Euler pole and employs a proper plate‐circuit argument to determine them numerically without resorting to approximations. The latter applies an involved interplay of incremental and accumulating modeling inducing split–join cycles to approximate sinusoidal trajectories as reported to record plates' motion during the Gondwana breakup. [ABSTRACT FROM AUTHOR]

Quaternion algebra is a noncommutative associative algebra. Noncommutativity limits the flexibility of computation and makes analysis related to quaternions nontrivial and challenging. Due to its applications in signal analysis and image processing, quaternionic Fourier analysis has received increasing attention in recent years. This paper addresses phase retrievability in quaternion Euclidean spaces H M . We obtain a sufficient condition on phase retrieval frames for quaternionic left Hilbert module ( H M , (· , ·)) of the form { e m T n g } m , n ∈ N M , where { e m } m ∈ N M is an orthonormal basis for ( H M , (· , ·)) and (· , ·) is the Euclidean inner product on H M . It is worth noting that { e m } m ∈ N M is not necessarily 1 M e 2 π i m · M m ∈ N M , and that our method also applies to phase retrievability in C M . For the real Hilbert space ( H M , ⟨ · , · ⟩) induced by ( H M , (· , ·)) , we present a sufficient condition on phase retrieval frames { e m T n g } m ∈ N 4 M , n ∈ N M , where { e m } m ∈ N 4 M is an orthonormal basis for ( H M , ⟨ · , · ⟩) . We also give a method to construct and verify general phase retrieval frames for ( H M , ⟨ · , · ⟩) . Finally, some examples are provided to illustrate the generality of our theory. [ABSTRACT FROM AUTHOR]

Abstract Nowadays a unit quaternion is widely employed to represent the three-dimensional (3D) rotation matrix and then applied to the 3D similarity coordinate transformation. A unit dual quaternion can describe not only the 3D rotation matrix but also the translation vector meanwhile. Thus it is of great potentiality to the 3D coordinate transformation. The paper constructs the 3D similarity coordinate transformation model based on the unit dual quaternion in the sense of errors-in-variables (EIV). By means of linearization by Taylor's formula, Lagrangian extremum principle with constraints, and iterative numerical technique, the Dual Quaternion Algorithm (DQA) of 3D coordinate transformation in weighted total least squares (WTLS) is proposed. The algorithm is capable to not only compute the transformation parameters but also estimate the full precision information of computed parameters. Two numerical experiments involving an actual geodetic datum transformation case and a simulated case from surface fitting are demonstrated. The results indicate that DQA is not sensitive to the initial values of parameters, and obtains the consistent values of transformation parameters with the quaternion algorithm (QA), regardless of the size of the rotation angles and no matter whether the relative errors of coordinates (pseudo-observations) are small or large. Moreover, the DQA is advantageous to the QA. The key advantage is the improvement of estimated precisions of transformation parameters, i.e. the average decrease percent of standard deviations is 18.28%, and biggest decrease percent is 99.36% for the scaled quaternion and translations in the geodetic datum transformation case. Another advantage is the DQA implements the computation and precision estimation of traditional seven transformation parameters (which still are frequent used yet) from dual quaternion, and even could perform the computation and precision estimation of the scaled quaternion. Graphical Abstract

This paper presents an investigation of brittle rock failure by the quaternion-based bonded-particle model in discrete element method (DEM). Unlike traditional approaches that utilize Euler angles or rotation matrices, this model employs unit quaternions to represent the spatial rotations of particles. This method simplifies the representation of 3D rotations, providing a more intuitive framework for modelling complex interactions in granular materials. The numerical model was validated by the uniaxial compression tests on rock, with good agreement with well-documented experimental data in terms of the rock uniaxial compression strength (UCS) and failure mode. During loading, the rock sample demonstrated a linear-elastic response at an axial strain of smaller than 0.45%. However, as internal bond breakage accumulated, this linear relationship weakened, and the stress-strain curve began to deviate from its initial linear trajectory. The bond breakage and the overall deformation of the rock were primarily controlled by the shear bonding force. The UCS was achieved at an axial strain of 0.625%, at which point the internal shear bonding force chains were predominantly aligned vertically. The brittle failure occurred when the internal damage of solids nucleated to form an interconnected failure plane, accompanied by a sharp rise in the internal damage ratio. The area of failure plane increased with the loading strain rate, gradually transforming the failure pattern from the local damage to a complete fragmentation.

With the wide application of quaternion, the solution of large quaternion structured matrix equations has become an important topic in scientific computing. Based on the new positive definite and skew-self-conjugate splitting iteration(NPSS), two new mixed-parameter splitting iteration schemes were constructed for the quaternion sub-positive definite system AX = B by introducing the two-parameter and relaxation acceleration techniques. That is, asymmetric new positive definite and skew-self-conjugate splitting ( ANPSS), successive over relaxation asymmetric new positive definite and skew-self-conjugate splitting ( SANPSS), and the convergence of these two iterations was proved by using the eigenvalue theory of quaternion matrix, and the range of relevant parameters was given. In addition, the complex representation method of quaternion matrix was used to realize the numerical solution of the system in MATLAB environment. Numerical examples show that the flexible selection of multiple parameters shows that the proposed mixed parameter splitting iteration has higher convergence efficiency than NPSS iteration. [ABSTRACT FROM AUTHOR]

In this paper, by introducing memristive connections and inertial term into quaternion‐valued neural networks, the quaternion‐valued inertial memristive neural networks (QVIMNNs) is proposed. Based on the adaptive control strategy, the synchronization problem of the QVIMNNs is analyzed in this work. Two different assumption for activation functions are considered in this work; new criteria for global synchronization are then derived by designing novel adaptive controllers and update laws. Lastly, numerical examples are given to verify the results obtained in this work. [ABSTRACT FROM AUTHOR]

The stability of nonlinear systems in the control domain has been extensively studied using different versions of the algebraic Riccati equation (ARE). This leads to the focus of this work: the search for the time-varying quaternion ARE (TQARE) Hermitian solution. The zeroing neural network (ZNN) method, which has shown significant success at solving time-varying problems, is used to do this. We present a novel ZNN model called 'ZQ-ARE' that effectively solves the TQARE by finding only Hermitian solutions. The model works quite effectively, as demonstrated by one application to quadrotor control and three simulation tests. Specifically, in three simulation tests, the ZQ-ARE model finds the TQARE Hermitian solution under various initial conditions, and we also demonstrate that the convergence rate of the solution can be adjusted. Furthermore, we show that adapting the ZQ-ARE solution to the state-dependent Riccati equation (SDRE) technique stabilizes a quadrotor's flight control system faster than the traditional differential-algebraic Riccati equation solution. [ABSTRACT FROM AUTHOR]

Let F be a real-closed field and D a F-division algebra. In this paper, we prove that if D is algebraic over F then the graph Γ(Mn(D)) is connected and its diameter is at most 4, for any n ≥ 3. If in addition, the division ring D is noncommutative, we also have the same results for n = 2. As a corollary, we show that the diameter of the commuting graph of the matrix algebra of degree n ≥ 2 over a generalized quaternion algebra ℍF(a,b), where F is a real-closed field, two elements a,b ∈ F∗, is also at most 4. This fact is a strong improvement of the previous result by Akbari et al. asserting that this diameter is at most 6 in case F is the field ℝ of real numbers. [ABSTRACT FROM AUTHOR]

Recent advances in practical fields like robotics and the miniaturisation of many sensors and actuators have turned autonomous vehicles into a reality. These systems come in many shapes and forms depending on the medium and their design but can be classified in one of two categories according to their mechanical configuration and number of actuators: fully-actuated and under-actuated vehicles; the second group still poses many challenges for the automatic control and robotics communities. This work introduces an easy methodology, based on quaternions, for designing controllers for a class of under-actuated systems, such as multicopter vehicles. The methodology starts from a fully-actuated representation of the system and adapts it to be implemented in under-actuated systems. In addition, it allows the easy development of controllers without taking care the coupled dynamics of the system. The proposed approach is based on Lyapunov's stability theory and is proven for certain criteria of the system's dynamics. The quadcopter configuration is used to validate the proposed methodology in simulations and in experimental tests. Abbreviations: AGV: autonomous ground vehicle; AUV: autonomous underwatervehicle; DoF: degree(s) of freedom; ESC: energy shaping controller; SFC: statefeedback controller; SSC: separated saturations controller; UAV: unmanned aerialvehicle [ABSTRACT FROM AUTHOR]

In this paper, we discuss the quaternion matrix equation A X ε + B X δ = 0 , where ε ∈ { I , C } , δ ∈ { † , ⁎ } and I , C , †, ⁎ denote the identity, involutive automorphism, involutive anti-automorphism and transpose, involutive automorphism and anti-automorphism and transpose, respectively. Firstly, we transform the given matrix equation into the matrix equation A ˜ Y ε + B ˜ Y δ = 0 with complex coefficient matrices and quaternion target matrix Y by utilizing the regularity of (A , B) , where A ˜ = P A Q and B ˜ = P B Q are two complex matrices with P , Q being two invertible quaternion matrices. Secondly, we show that the solution can be obtained in terms of Q , the Kronecker canonical form of (A ˜ , B ˜) and one of the two invertible quaternion matrices which transform (A ˜ , B ˜) into its Kronecker canonical form. Meanwhile, we also decouple the transformed equation into some systems of small-scale equations in terms of Kronecker canonical form of (A ˜ , B ˜). Thirdly, we determine the dimension of the solution space of the equation in terms of the sizes of the blocks arising in the Kronecker canonical form. Finally, we give the necessary and sufficient condition for the existence of the unique solution. [ABSTRACT FROM AUTHOR]

Keeping in view the advantages of quaternion, an attempt is made to explain supersymmetry breaking in the interacting field in terms of quaternion superpartner Hamiltonian. A complete theory for supersymmetry breaking in the thermal field in terms of bosonic and fermionic operators has been obtained along with certain restrictions due to the non-commutativity of quaternions that can hint at some hidden results. Witten index in terms of quaternion units has been obtained, and we find that energy is different along different axes. It has been observed that supersymmetry remains good symmetry at zero temperature but breaks down with temperature. [ABSTRACT FROM AUTHOR]

A knowledge graph is a structured semantic network designed to describe physical entities and relations in the world. A comprehensive and accurate knowledge graph is essential for tasks such as knowledge inference and recommendation systems, making link prediction a popular problem for knowledge graph completion. However, existing approaches struggle to model complex relations among entities, which severely hampers their ability to complete knowledge graphs effectively. To address this challenge, we propose a novel hierarchical multi-head attention network embedding framework, called RiQ-KGC, which integrates different-grained contextual information of knowledge graph triples and models quaternion rotation relations between entities. Furthermore, we propose a relation instantiation method for alleviating the difficulty of expressing complex relations between entities. To enhance the expressiveness of relation representation, the relation is integrated by Transformer to obtain multi-hop neighbor information, so that one relation can be embedded into different embeddings according to different entities. Experimental results on four datasets demonstrate that RiQ-KGC exhibits strong competitiveness compared to state-of-the-art models in link prediction, while the ablation experiments reveal that the proposed relation instantiation method achieves great performance. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the S-eigenvalue of finite type of a bounded right quaternionic linear operator acting in a right quaternionic Hilbert space. The study is based on the different properties of the Riesz projection associated with the connected part of the S-spectrum. Furthermore, we introduce the left and right Browder S-resolvent operators. Inspired by the S-resolvent equation, we give the Browder’s S-resolvent equation in quaternionic setting. [ABSTRACT FROM AUTHOR]

As one promising solution, zero-watermarking approaches have been proposed to enhance the image visual quality and applied to protect the intellectual property rights of the medical images, remote sensing images and military images. The moments and their invariants have good image descriptive capability and geometrical invariance, which are common tools for image zero-watermarking. However, there are still some problems. Firstly, most of the existing algorithms ignore the problem of balancing robustness and discriminability. Secondly, the direct computation of image moments is time-consuming, numerically inaccurate, and unstable, thus affecting the effectiveness of these moment-based methods. Third, there is less research on zero-watermarking algorithms targeted at color images. To address these problems, this paper presents a zero-watermarking algorithm which is based on accurate quaternion fractional-order pseudo-Jacobi–Fourier moments with time–frequency analysis capability for color images. Firstly, we define a new set of orthogonal moments, named fractional-order pseudo-Jacobi–Fourier moments (FPJFMs), which is characterized by the generic nature and time–frequency analysis capability. And, we propose a scheme for the accurate computation of FPJFMs. Also, we extend accurate FPJFMs to the accurate quaternion FPJFMs (QFPJFMs), which is an efficient color image representation. Next, a new framework is proposed, namely hybrid low-order moment feature (HLMF) based on accurate QFPJFMs, which can simultaneously improve the legibility and robustness of the image representation. Finally, the accurate QFPJFMs-based HLMF is adapted to the zero-watermarking algorithm. Numerous experiment results indicate that this paper's presented zero-watermarking approach provides a good balance between robustness and discriminability, which is an effective and superior scheme. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Ordnance Equipment Engineering is the property of Chongqing University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we show that the set O/Zp, where p is a prime number, does not form a skew field and discuss idempotent and nilpotent elements in the (finite) ring O/Zp. We provide examples and establish conditions for idempotency and nilpotency. [ABSTRACT FROM AUTHOR]

Dynamic Sylvester equation (DSE) problems have drawn a lot of interest from academics due to its importance in science and engineering. Due to this, the quest for the quaternion DSE (QDSE) solution is the subject of this work. This is accomplished using the zeroing neural network (ZNN) technique, which has achieved considerable success in tackling time-varying issues. Keeping in mind that the original ZNN can handle QDSE successfully in a noise-free environment, but it might not work in a noisy one, and the noise-resilient ZNN (NZNN) technique is also utilized. In light of that, one new ZNN model is introduced to solve the QDSE problem and one new NZNN model is introduced to solve the QDSE problem under different types of noises. Two simulation experiments and one application to control of the sine function memristor (SFM) chaotic system show that the models function superbly.

Due to its significance in science and engineering, time-varying linear matrix equation (LME) problems have received a lot of attention from scholars. It is for this reason that the issue of finding the minimum-norm least-squares solution of the time-varying quaternion LME (ML-TQ-LME) is addressed in this study. This is accomplished using the zeroing neural network (ZNN) technique, which has achieved considerable success in tackling time-varying issues. In light of that, two new ZNN models are introduced to solve the ML-TQ-LME problem for time-varying quaternion matrices of arbitrary dimension. Two simulation experiments and two practical acoustic source tracking applications show that the models function superbly.

In the 21st century, computer technology covers human daily life, including the field of music. This paper uses a quaternion algorithm to describe the gesture posture during piano playing. Combined with the iteratively updated and extended IU-EKF algorithm, it realizes the fusion of piano playing gestures to fix the posture. The recognized piano playing gestures are output to the chord fingering automatic annotation board, the piano audio signal data are preprocessed and input to the 3D space, the annotation area is allocated, and the chord fingering features are extracted using the Boltzmann machine. Through spectral analysis and empirical investigation, we analyze the sound quality effect of the two pianos and the audience’s listening experience. The results show that in the spectral analysis of the two pianos, the time-domain waveforms of the pianos played using the gestures proposed in this paper have durations ranging from 0.75s to 1.25s, and the waveform graphs present triangular shapes, which are better in terms of sound quality. The listeners’ melodic memory of the songs played by two pianos is the best for the two pianos of Chinese art and folk songs, Chinese art and Chinese pop, with the average difference of 5.7899 and 5.6345 respectively. The two pianos form of playing can satisfy the listener’s need of listening to the piano songs to a certain extent.

Based on hierarchical inner-outer loop strategy, the tracking control for the helicopter system could be designed individually for the position loop and for the attitude loop, thus simplifying the underactuated control problem. However, due to the nonlinear coupling between the position dynamics and rotation dynamics, the performance of the position control is affected by attitude errors, especially when the attitude control can not tracks the reference attitude instantaneously. This work provides a hierarchical trajectory tracking control design for the helicopter with model uncertainties, ensuring the stability of the overall system considering the perturbation caused by attitude tracking errors and the nonlinear coupling. The attitude of the helicopter is descried by unit-quaternion, for which anti-unwinding control design is presented. Besides, the criteria for avoidance of singularity in generation of the reference attitude is derived. Simulation results demonstrate the effectiveness of the design. [ABSTRACT FROM AUTHOR]

In recent years, it has become essential to compute the time-varying quaternion (TVQ) matrix Moore-Penrose inverse (MP-inverse or pseudoinverse) to solve time-varying issues in a range of disciplines, including engineering, physics and computer science. This study examines the problem of computing the TVQ matrix MP-inverse using the zeroing neural network (ZNN) approach, which is nowadays considered a cutting edge technique. As a consequence, three new ZNN models are introduced for computing the TVQ matrix MP-inverse in the literature for the first time. Particularly, one model directly employs the TVQ input matrix in the quaternion domain, while the other two models, respectively, use its complex and real representations. In four numerical simulations and a real-world application involving robotic motion tracking, the models exhibit excellent performance.

The quaternion was introduced via the special unitary group in two dimensions and its properties were discussed systematically and rigorously. Some important results were educed such as the homomorphism of SU（2） onto SO（3）. Besides， the axis and the angle of the rotation were given by a quaternion， and the angular displacement of an infinitesimal rotation was derived smoothly.

A knowledge graph is a structured semantic network designed to describe physical entities and relations in the world. A comprehensive and accurate knowledge graph is essential for tasks such as knowledge inference and recommendation systems, making link prediction a popular problem for knowledge graph completion. However, existing approaches struggle to model complex relations among entities, which severely hampers their ability to complete knowledge graphs effectively. To address this challenge, we propose a novel hierarchical multi-head attention network embedding framework, called RiQ-KGC, which integrates different-grained contextual information of knowledge graph triples and models quaternion rotation relations between entities. Furthermore, we propose a relation instantiation method for alleviating the difficulty of expressing complex relations between entities. To enhance the expressiveness of relation representation, the relation is integrated by Transformer to obtain multi-hop neighbor information, so that one relation can be embedded into different embeddings according to different entities. Experimental results on four datasets demonstrate that RiQ-KGC exhibits strong competitiveness compared to state-of-the-art models in link prediction, while the ablation experiments reveal that the proposed relation instantiation method achieves great performance.

The asymptotic synchronization of quaternion-valued delayed neural networks with impulses and inertia is studied in this article. Firstly, a convergence result on piecewise differentiable functions is developed, which is a generalization of the Barbalat lemma and provides a powerful tool for the convergence analysis of discontinuous systems. To achieve synchronization, a constant gain-based control scheme and an adaptive gain-based control strategy are directly proposed for response quaternion-valued models. In the convergence analysis, a direct analysis method is developed to discuss the synchronization without using the separation technique or reduced-order transformation. In particular, some Lyapunov functionals, composed of the state variables and their derivatives, are directly constructed and some synchronization criteria represented by matrix inequalities are obtained based on quaternion theory. Some numerical results are shown to further confirm the theoretical analysis.

Let Q = (a,b / R) denote the quaternion algebra over the reals which is by the Frobenius Theorem either split or the division algebra H of Hamilton's quaternions. We have presented explicitly in [4] the matrix of a typical derivation of Q. Given a derivation d ∈ Der(H), we show that the matrix D ∈ Ma (ℝ) that represents d on the linear subspace ... of pure quaternions provides a pair of idempotent matrices AdjD and - D² that correspond bijectively to the involutary matrix Ʃof a quaternion involution σ and present several equations involving these matrices. In particular, we deal with commuting derivations of H and introduce some results to guarantee commutativity. We also mention briefly eigenspace decomposition of a derivation. [ABSTRACT FROM AUTHOR]

Based on the quaternion system, we give a new representation of the complex vague soft set, and related logical operations. This new representation contains more information than before. Three quaternion distance measures are proposed and a decision model is established. The disease diagnosis of breast cancer is applied to the model to reflect the superiority of the model. By comparing the diagnostic errors under the different distance measures, the most suitable distance measure for this dataset is selected. [ABSTRACT FROM AUTHOR]

This paper discuss necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quadratic matrix equation X²+BX+XB*+Q=0. on the quaternion field and its iterative solution method. Some necessary and sufficient conditions for the existence of Hermite positive definite solutions of this equation are proved mainly for the characteristics of coefficient matrices by introducing appropriate parameters to establish matrix inequalities and by using the theory of fixed points on convex sets. On this basis, three convergent iteration formats are constructed for different conditions and solution existence intervals, and the selection of the initial matrix is given according to each iteration property. The algorithm is solved in the Matlab environment by using the complexization operator of the quaternion matrix. At the same time a perturbation analysis is carried out on the solution of the equation, and two perturbation error bounds are obtained. Three numerical examples are used to test the effectiveness and feasibility of the given method. [ABSTRACT FROM AUTHOR]

We study the Rayleigh quotient of a Hermitian matrix with quaternionic coefficients and prove its main properties. As an application, we give some relationships between left and right eigenvalues of Hermitian and symplectic matrices. [ABSTRACT FROM AUTHOR]

This work introduces fractional d-bar derivatives in the setting of quaternionic analysis, by giving meaning to fractional powers of the quaternionic d-bar derivative. The definition is motivated by starting from nth-order d-bar derivatives for n ∈ N , and further justified by various natural properties such as composition laws and its action on special functions such as Fueter polynomials. [ABSTRACT FROM AUTHOR]

The stochastic disturbances are common in real world and usually cause significant influence to engineering system. In this work, the stochastic disturbance is introduced into the quaternion-valued memristive neural networks (QVMNNs). The exponential input-to-state stabilization (EITSS) problem of stochastic QVMNNs is investigated. In order to be more effective and less costly in real applications, an event-triggered control strategy is adopted. The original QVMNNs are separated into four equivalent real-valued NNs by using Hamilton rule. Then, by using the Lyapunov functional approach and stochastic analysis technique, novel sufficient conditions for mean square EITSS of stochastic QVMNNs are derived. Moreover, it is proved that Zeno behavior will not take place in our event-triggered control method. Thus, the mean square EITSS problem of stochastic QVMNNs is solved in this work with less control cost. Lastly, simulation is performed to manifest the correctness of the theorem. [ABSTRACT FROM AUTHOR]