Quaternion Multiplication Research Articles

Transquaternions

The quaternions extend the complex numbers and are used in physics and engineering. Division of quaternions by zero is not defined, which limits physical theories and engineering applications. We now introduce transquaternions, which totalise the arithmetical operations of quaternion addition, subtraction, multiplication, and both left and right division. In particular, division of quaternions by zero is allowed. The transquaternions are homeomorphic to the unit hypersphere or glome, including its interior, together with an isolated point. The 4D interior of the hypersphere is made up of the ordinary quaternions. The 3D surface of the hypersphere is made up of the infinite transquaternions, which are produced by dividing non-zero quaternions by zero. The isolated point, that lies outside the 4D space containing the hypersphere, is the transquaternion nullity, which is produced by dividing zero by zero. Transquaternions are a separable compact complete metric topological space.

Global Dissipativity Analysis and Stability Analysis for Fractional-Order Quaternion-Valued Neural Networks With Time Delays

This article studies dissipativity analysis of fractional-order quaternion-valued neural networks (FOQVNNs) with time delays. Two specific activation functions are considered along with common bounded and activation functions of Lipschitz-kind. Since quaternion multiplication is not commutative, we must divide the model, which is evaluated by quaternion, into four elements that are real-valued elements. On the basis of the construction of novel Lyapunov functional, and applying fractional-calculus theory, new criteria for the test of the global dissipativity and exponential stability of FOQVNNs model are established. FOQVNNs have also been suggested to provide global dissipativity and exponential stability, whereas nonlinear complex activation functions are constrained by the usage of linear matrix inequality methods, which utilize quaternion matrices and positive quaternion definite matrices. Finally, the effectiveness and superiority of the proposed approach is validated through numerical examples.

Zeroing Neural Networks for Dynamic Quaternion-Valued Matrix Inversion

This article, for the first time, extends the zeroing neural network (ZNN) method to address the problem of dynamic quaternion-valued matrix inversion. Due to the noncommutative property of quaternion multiplication, the complex representation method is first adopted to transform quaternion-valued matrices into the corresponding complex-valued matrices. Then, based on two kinds of ways to deal with nonlinear activation functions in the complex-valued domain, this article proposes two quaternion-valued ZNN (QVZNN) models for dynamic quaternion-valued matrix inversion. In addition, a novel nonlinear activation function is given to accelerate the convergence rate of the models to reach the predefined-time convergence. The detailed theoretical analysis, together with four theorems, are given to show the excellent properties of the QVZNN models. Furthermore, the upper bound of the convergence time is derived analytically with the residual error being zero theoretically. Finally, two numerical examples are provided to verify the theoretical results and the effectiveness of the QVZNN models for the dynamic quaternion-valued matrix inversion, and an application to mobile manipulator control is provided to indicate the practical application value of the QVZNN models.

Stochastic stability criteria and event-triggered control of delayed Markovian jump quaternion-valued neural networks

This paper investigates, through designing an event-triggered sampled-data mechanism, the stochastic stabilization of Markovian jump quaternion-valued neural networks (QVNNs) with partially unknown transition probabilities. First, since the multiplication of quaternions is not commutative, the Markovian jump QVNNs are decomposed into four real-valued neural networks. Then, by constructing a novel augmented Lyapunov–Krasovskii functional (LKF), it is containing more hybrid features, in which concerning the practical sampling pattern, and using the convex combinatorial optimization and inequality technique, the stabilization condition is proposed in the form of linear matrix inequalities (LMIs). Furthermore, the desired event-triggered sampled-data controller gains are obtained for the closed-loop Markovian jump QVNNs with partially unknown transition probability. Finally, the viability and validness of the obtained results are confirmed via three numerical examples.

Dual Quaternion Hardware Accelerator for RISC-V based System

In this work, a hardware accelerator has been developed for a RISC-V processor. The ‘Parashu’ Shakti processor is the SoC of choice for application testing and development. The IP is designed to speed up applications involving dual quaternion operations. Our module primarily aids dual quaternion multiplication which further helps with other complex operations like translation, rotation and transformation. Two solutions have been proposed for the same, i.e. either a quaternion IP if power and resource utilization is a concern, or a dual quaternion IP if performance gain is the primary objective. The latter is however at the expense of relatively more resource utilization. The former IP takes longer execution time to perform the same task but is more versatile since it can be used in applications involving both quaternion and dual quaternion operations.

Dynamical degrees of automorphisms on abelian varieties

For any given Salem number, we construct an automorphism on a simple abelian variety whose first dynamical degree is the square of the Salem number. Our construction works for both simple abelian varieties with totally indefinite quaternion multiplication and for simple abelian varieties of the second kind. We then give a complete classification of the dynamical degree sequences for abelian varieties of dimension at most four and obtain an ergodic result for sequences of pullbacks of forms.

Global exponential synchronization of high-order quaternion Hopfield neural networks with unbounded distributed delays and time-varying discrete delays

In this paper, the synchronization control problem of high-order quaternion Hopfield neural networks (HOQHNNs) with unbounded distributed delays and time-varying discrete delays is studied. Since the non-exchangeability of quaternion multiplication operation, the error system is first decomposed into equivalent real-valued systems, and then the driving-response structure of HOQHNN is achieved exponential synchronization by designing a state feedback controller. At the same time, two examples are offered to illustrate the applicability of the consequences obtained. Finally, the obtained global exponential synchronization criterion is applied to encrypt and decrypt color images. The method proposed here has four advantages: (1) There is no need to construct any Lyapunov–Krasovskii functional, which is straight in the light of the definition of global exponential synchronization; (2) The obtained synchronization criterion contains only a few simple inequalities, which can find the solutions readily; (3) The proposed method is available to real-valued and complex-valued neural networks; (4) The obtained results are helpful for the image secure transmission.

Fixed point free actions of spheres and equivariant maps

The concept of index and co-index of a paracompact Hausdorff space X equipped with free involutions relative to the antipodal action on spheres were introduced by Conner and Floyd [2]. In this paper, we extend the notion of index and co-index for free G-spaces X, where X is a finitistic space and G=S1 (with complex multiplication) and G=S3 (with quaternionic multiplication). We prove that the index of X is less than or equal to the mod 2 cohomology index of X. We also compute the ring cohomology of the orbit space X/G, where G=S1 or G=S3 and X is a finitistic space whose cohomology ring is the same as the product of spheres Sn×Sm,1≤n≤m. Using these cohomological calculations, we obtain some Borsuk-Ulam type results.

Quantitative Reduction Theory and Unlikely Intersections

AbstractWe prove quantitative versions of Borel and Harish-Chandra’s theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive group. Secondly, we obtain polynomial bounds in the construction of fundamental sets for arithmetic subgroups of reductive groups, as the latter vary in a real conjugacy class of subgroups of a fixed reductive group. Our results allow us to apply the Pila–Zannier strategy to the Zilber–Pink conjecture for the moduli space of principally polarised abelian surfaces. Building on our previous paper, we prove this conjecture under a Galois orbits hypothesis. Finally, we establish the Galois orbits hypothesis for points corresponding to abelian surfaces with quaternionic multiplication, under certain geometric conditions.

Stochastic stabilization of Markov jump quaternion-valued neural network using sampled-data control

This paper investigates the stochastic stabilization of Markov jump quaternion-valued neural networks (QVNNs) using a sampled-data control strategy. Firstly, Markov jump QVNNs are decomposed into two complex-valued systems using the plural decomposition method because the multiplication of quaternions is not commutative. Secondly, the existence and uniqueness of the equilibrium point of the Markov jump QVNNs is proved according to the theory of homeomorphism mapping. Thirdly, by choosing a suitable Lyapunov-Krasovskii functional and combining some inequality techniques, a new stochastic stability criterion is established for the Markov jump QVNNs. Based on this, several verifiable sufficient conditions for the stochastic stabilization of Markov jump QVNNs with sampled-data control are ensured. Finally, the correctness and effectiveness of the proposed method are verified by two numerical examples.

Fast Algorithms for Quaternion-Valued Convolutional Neural Networks.

In this article, we analyze algorithmic ways to reduce the arithmetic complexity of calculating quaternion-valued linear convolution and also synthesize a new algorithm for calculating this convolution. During the synthesis of the discussed algorithm, we use the fact that quaternion multiplication may be represented as a matrix-vector product. The matrix participating in the product has unique structural properties that allow performing its advantageous decomposition. Namely, this decomposition leads to reducing the multiplicative complexity of computations. In addition, we used the fact that when calculating the elements of the quaternion-valued convolution, the part of the calculations of all matrix-vector products is common. It gives an additional reduction in the number of additions of real numbers and, consequently, a decrease in the additive complexity of calculations. Thus, the use of the proposed algorithm will contribute to the acceleration of calculations in quaternion-valued convolution neural networks.

Attitude Estimation Using Low-Cost MARG Sensors With Disturbances Reduction

Attitude information obtained from low-cost magnetic, angular rate, and gravity (MARG) sensors is prone to be deteriorated by disturbances such as external acceleration and magnetic anomaly in operational environments. To address the problem, a robust attitude estimation algorithm using low-cost MARG sensors is proposed. An error state Kalman filter (ESKF) is used for data fusing, in which the attitude errors are interpreted as a small rotation vector, and the gyro bias variation is also estimated. The nominal state and the error state are propagated with unit quaternion and rotation vector, respectively. An attitude correction is followed which combines the nominal and error parts by quaternion multiplication. The robustness of the proposed algorithm is embodied in resisting accuracy degradation of the attitude estimation caused by both external acceleration and magnetic anomalies. While external accelerations and magnetic disturbances are detected online, the accuracy reduction in attitude estimation is prevented by adaptively adjusting the related measurement noise covariance matrix. Besides, a two-step measurement update strategy is designed to guarantee that the roll and pitch update is separated from the yaw update. Various rotation and land vehicle dynamic tests have been conducted to validate the effectiveness and robustness of the proposed algorithm.

Robust Sparse Representation in Quaternion Space.

Sparse representation has achieved great success across various fields including signal processing, machine learning and computer vision. However, most existing sparse representation methods are confined to the real valued data. This largely limit their applicability to the quaternion valued data, which has been widely used in numerous applications such as color image processing. Another critical issue is that their performance may be severely hampered due to the data noise or outliers in practice. To tackle the problems above, in this work we propose a robust quaternion valued sparse representation (RQVSR) method in a fully quaternion valued setting. To handle the quaternion noises, we first define a new robust estimator referred as quaternion Welsch estimator to measure the quaternion residual error. Compared to the conventional quaternion mean square error, it can largely suppress the impact of large data corruption and outliers. To implement RQVSR, we have overcome the difficulties raised by the noncommutativity of quaternion multiplication and developed an effective algorithm by leveraging the half-quadratic theory and the alternating direction method of multipliers framework. The experimental results show the effectiveness and robustness of the proposed method for quaternion sparse signal recovery and color image reconstruction.

Finite-time synchronization of delayed fractional-order quaternion-valued memristor-based neural networks

This paper mainly concerns with the finite-time synchronization of delayed fractional-order quaternion-valued memristor-based neural networks (FQVMNNs). First, the FQVMNNs are studied by separating the system into four real-valued parts owing to the noncommutativity of quaternion multiplication. Then, two state feedback control schemes, which include linear part and discontinuous part, are designed to guarantee that the synchronization of the studied networks can be achieved in finite time. Meanwhile, in terms of the stability theorem of delayed fractional-order systems, Razumikhin technique and comparison principle, some novel criteria are derived to confirm the synchronization of the studied models. Furthermore, two methods are used to obtain the estimation bounds of settling time. Finally, the feasiblity of the synchronization methods in quaternion domain is validated by the numerical examples.

Relativistic Combination of Non-Collinear 3-Velocities Using Quaternions

Quaternions have an (over a century-old) extensive and quite complicated interaction with special relativity. Since quaternions are intrinsically 4-dimensional, and do such a good job of handling 3-dimensional rotations, the hope has always been that the use of quaternions would simplify some of the algebra of the Lorentz transformations. Herein we report a new and relatively nice result for the relativistic combination of non-collinear 3-velocities. We work with the relativistic half-velocities w defined by v=2w1+w2, so that w=v1+1−v2=v2+O(v3), and promote them to quaternions using w=wn^, where n^ is a unit quaternion. We shall first show that the composition of relativistic half-velocities is given by w1⊕2≡w1⊕w2≡(1−w1w2)−1(w1+w2), and then show that this is also equivalent to w1⊕2=(w1+w2)(1−w2w1)−1. Here as usual we adopt units where the speed of light is set to unity. Note that all of the complicated angular dependence for relativistic combination of non-collinear 3-velocities is now encoded in the quaternion multiplication of w1 with w2. This result can furthermore be extended to obtain novel elegant and compact formulae for both the associated Wigner angle Ω and the direction of the combined velocities: eΩ=eΩΩ^=(1−w1w2)−1(1−w2w1), and w^1⊕2=eΩ/2w1+w2|w1+w2|. Finally, we use this formalism to investigate the conditions under which the relativistic composition of 3-velocities is associative. Thus, we would argue, many key results that are ultimately due to the non-commutativity of non-collinear boosts can be easily rephrased in terms of the non-commutative algebra of quaternions.

Picard–Fuchs equations for Shimura curves over Q

We show that Picard-Fuchs equations of periods of certain families of abelian surfaces with quaternionic multiplication have fractional powers of algebraic modular forms as solutions. We give sever ...

Quaternion block sparse representation for signal recovery and classification

This paper presents a quaternion block sparse representation (QBSR) method for structural sparse signal recovery in quaternion space. Due to the noncommutativity of quaternion multiplication, conventional optimization algorithms originally designed for real-valued and complex-valued signal recovery problems are not applicable to the optimization problem of QBSR. To combat this problem, we leverage several quaternion operators and devise an effective algorithm for QBSR within the ADMM (Alternating Direction Method of Multipliers) framework. The second contribution of this work is to develop a QBSR based classifier referred to as QBSRC for quaternion data classification with application to color face recognition. Compared with real-valued representation based classifiers handling multiple color channels of a color image independently, QBSRC treats a color image as a quaternion signal and represents it in a holistic manner. The third contribution is to provide the theoretical analysis of QBSRC and rigorously prove that QBSRC is guaranteed to succeed in classification of any new test sample under appropriate condition. Experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed method for quaternion signal recovery and classification.

Impulsive disturbance on stability analysis of delayed quaternion-valued neural networks

In reality, abrupt uncertainty phenomenon may disturb or even break the system stability. Under these circumstances, it is worthy of studying the stability criteria of impulsive disturbed systems. In this article, an impulsive disturbed neural network model with delays is constructed in quaternion space, and the exponential stability conditions of the delayed system are derived by utilizing generalized norms. Firstly, a general impulsive disturbed quaternion-valued delayed neural network (IQVDNN) is given by combining impulsive differential system with quaternion-valued neural networks. Since quaternion multiplication is noncommutative, the IQVDNN system is decomposed into real-valued impulsive delayed neural networks. Then, the generalized ∞-norm and 1-norm are used to research its stability, respectively. By constructing special Lyapunov-type functional, several exponential stability sufficient criteria are obtained, which can guarantee that the stability can’t be destroyed by abrupt impulsive perturbations. Finally, two numerical examples and their simulation figures are given to show the effectiveness of the obtained conclusions.

Novel Inequalities to Global Mittag-Leffler Synchronization and Stability Analysis of Fractional-Order Quaternion-Valued Neural Networks.

This article is concerned with the problem of the global Mittag-Leffler synchronization and stability for fractional-order quaternion-valued neural networks (FOQVNNs). The systems of FOQVNNs, which contain either general activation functions or linear threshold ones, are successfully established. Meanwhile, two distinct methods, such as separation and nonseparation, have been employed to solve the transformation of the studied systems of FOQVNNs, which dissatisfy the commutativity of quaternion multiplication. Moreover, two novel inequalities are deduced based on the general parameters. Compared with the existing inequalities, the new inequalities have their unique superiorities because they can make full use of the additional parameters. Due to the Lyapunov theory, two novel Lyapunov-Krasovskii functionals (LKFs) can be easily constructed. The novelty of LKFs comes from a wider range of parameters, which can be involved in the construction of LKFs. Furthermore, mainly based on the new inequalities and LKFs, more multiple and more flexible criteria are efficiently obtained for the discussed problem. Finally, four numerical examples are given to demonstrate the related effectiveness and availability of the derived criteria.

Global Mittag-Leffler Boundedness of Fractional-Order Fuzzy Quaternion-Valued Neural Networks With Linear Threshold Neurons

This article focuses on the global Mittag-Leffler boundedness for fractional-order fuzzy quaternion-valued neural networks (QVNNs) with linear threshold neurons. In order to avert the nonexchangeability for quaternion multiplication, the considered QVNN is separated into four real-valued or two complex-valued systems. By employing Lyapunov function method and fractional-order differential inequalities, several effective conditions according to algebraic inequality and complex-valued linear matrix inequalities are deduced to guarantee the global Mittag-Leffler boundedness of the addressed network. And, the frame of the global Mittag-Leffler attracting sets is presented as well. Here, the numbers of the equilibrium points need not to be concerned. Finally, a numerical example is provided to demonstrate the correctness of the proposed results.