Quaternion Multiplication Research Articles

Exponential input-to-state stability of quaternion-valued neural networks with time delay

This paper debated the exponential input-to-state stability (EITSS) of the solution for a kind of quaternion-valued neural networks (QVNNs) with time delay. It fills the blank of QVNN in the aspect of input-to-state stability (ITSS). In virtue of the quaternion multiplication is not suitable for commutative law, QVNN is ordinarily resolved into four real-valued neural networks (RVNNs). Making use of a novel Lyapunov–Krasovskii function and some inequalities, we obtain a little sufficient conditions to assure the considered system is EITSS. Finally, by means of two examples, it is certified that the calculation results in this paper are fine.

The Shimura canonical model for the quaternion algebra of discriminant 6 on the Ihara-Kurihara conic

In this paper we construct the Shimura canonical model for the quaternion algebra B of discriminant 6 over Q in the sense of the original paper [SmrB] (see also [Shg2]). The “unit group” of B becomes to be a quadrangle group Γ=□(0;2,2,3,3) acting on the complex upper half space H. For our purpose we use the defining equation of the Shimura curve V=H/□(0;2,2,3,3) given by Y. Ihara which found in the work of A. Kurihara (1979). We are requested to find a modular function ψ defined on H with respect to Γ which realizes V satisfying several arithmetic conditions stated later. Basically our ψ is the modular function for the family of abelian surfaces with quaternion multiplication by B, so called false elliptic curves, with some arithmetic conditions. Our main tool is the hypergeometric modular function given by M. Petkova and H. Shiga (2011). This is the full paper of the survey report [Shg2] (2018).

Stability Analysis of Quaternion-Valued Neutral-Type Neural Networks with Time-Varying Delay

This paper addresses the problem of global μ -stability for quaternion-valued neutral-type neural networks (QVNTNNs) with time-varying delays. First, QVNTNNs are transformed into two complex-valued systems by using a transformation to reduce the complexity of the computation generated by the non-commutativity of quaternion multiplication. A new convex inequality in a complex field is introduced. In what follows, the condition for the existence and uniqueness of the equilibrium point is primarily obtained by the homeomorphism theory. Next, the global stability conditions of the complex-valued systems are provided by constructing a novel Lyapunov–Krasovskii functional, using an integral inequality technique, and reciprocal convex combination approach. The gained global μ -stability conditions can be divided into three different kinds of stability forms by varying the positive continuous function μ ( t ) . Finally, three reliable examples and a simulation are given to display the effectiveness of the proposed methods.

Examples of genuine QM abelian surfaces which are modular

Let K be an imaginary quadratic field. Modular forms for GL(2) over K are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over K or an associated abelian surface with quaternionic multiplication over K. We give explicit evidence in the way of examples to support this conjecture in the latter case. Furthermore, the quaternionic surfaces given correspond to genuine Bianchi newforms, which answers a question posed by Cremona as to whether this phenomenon can happen.

Effects of State-Dependent Impulses on Robust Exponential Stability of Quaternion-Valued Neural Networks Under Parametric Uncertainty.

This paper addresses the state-dependent impulsive effects on robust exponential stability of quaternion-valued neural networks (QVNNs) with parametric uncertainties. In view of the noncommutativity of quaternion multiplication, we have to separate the concerned quaternion-valued models into four real-valued parts. Then, several assumptions ensuring every solution of the separated state-dependent impulsive neural networks intersects each of the discontinuous surface exactly once are proposed. In the meantime, by applying the B -equivalent method, the addressed state-dependent impulsive models are reduced to fixed-time ones, and the latter can be regarded as the comparative systems of the former. For the subsequent analysis, we proposed a novel norm inequality of block matrix, which can be utilized to analyze the same stability properties of the separated state-dependent impulsive models and the reduced ones efficaciously. Afterward, several sufficient conditions are well presented to guarantee the robust exponential stability of the origin of the considered models; it is worth mentioning that two cases of addressed models are analyzed concretely, that is, models with exponential stable continuous subsystems and destabilizing impulses, and models with unstable continuous subsystems and stabilizing impulses. In addition, an application case corresponding to the stability problem of models with unstable continuous subsystems and stabilizing impulses for state-dependent impulse control to robust exponential synchronization of QVNNs is considered summarily. Finally, some numerical examples are proffered to illustrate the effectiveness and correctness of the obtained results.

Stability analysis of fractional Quaternion-Valued Leaky Integrator Echo State Neural Networks with multiple time-varying delays

This paper studies the stability problem of fractional order continuous-time Quaternion-Valued Leaky Integrator Echo State Neural Networks (NN) with multiple time-varying delays. The NN is decomposed into four real-valued systems without considering the non-commutativity of quaternion multiplication resulting in Hamilton rules. Then, the delay-independent stability analysis of the NN equilibrium point with Lipschitz continuous activation function is derived in the real space. The existence and uniqueness of the equilibrium point are also given using the contraction map in the real space. Simulations results show the feasibility of the method.

Eigen-Decomposition of Quaternions

This paper introduces biquaternion eigen-decomposition theory (via Peirce decomposition) with respect to a selected quaternion with a non-zero vector part. The eigen-decomposition allows evaluation of polynomials and power series with real coefficients as functions of quaternions. This extension of analytic functions to functions of quaternions requires only standard complex function evaluation. The theory also applies to quaternion rotations. The theory uses biquaternion calculations indicated by matrix methods via the algebraic isomorphism between Hamilton’s biquaternions and appropriate $$4\times 4$$ complex matrices. The isomorphism preserves algebraic structure. In particular, the left and right biquaternion multiplication by the selected quaternion maps to left and right matrix multiplication, respectively. This unifies the representation of the left and right quaternion multiplication as a linear map into a single matrix form. This matrix, as a linear operator, acts on matrices, so that the eigenvectors have matrix form that maps into the biquaternions. Use of an alternate quaternion basis results in a similarity transform of the representation matrix, preserving eigenvalues across change of basis. The similarity transform allows simple eigenvector calculation. The matrix for the selected quaternion has two identical, complex conjugate pairs of eigenvalues. Each pair corresponds to two complex conjugate pairs of eigenvector biquaternions, an idempotent pair and a nilpotent pair. Idempotent and nilpotent eigenvectors correspond to the commuting and non-commuting parts, respectively, of quaternion multiplication.

Multistability analysis of delayed quaternion-valued neural networks with nonmonotonic piecewise nonlinear activation functions

This paper deals with the multistability problem of the quaternion-valued neural networks (QVNNs) with nonmonotonic piecewise nonlinear activation functions and unbounded time-varying delays. By virtue of the non-commutativity of quaternion multiplication resulting from Hamilton rules, the QVNNs can be separated into four real-valued systems. By using the fixed point theorem and other analytical tools, some novel algebraic criteria are established to guarantee that the QVNNs can have 54n equilibrium points, 34n of which are locally μ-stable. Some criteria that guarantee the multiple exponential stability, multiple power stability, multiple log-stability, multiple log–log-stability are also derived as special cases. The obtained results reveal that the introduced QVNNs in this paper can have larger storage capacity than the complex-valued ones. Finally, one numerical example is presented to clarify the validity of the theoretical results.

Twin-multistate commutative quaternion Hopfield neural networks

Abstract The complex-valued Hopfield neural network (CHNN) can deal with multi-level information, and has often been applied to the storage of image data. The quaternion Hopfield neural network (QHNN) is a multistate model of a Hopfield neural network, and requires half the connection weight parameters of CHNN. In this study, we propose a commutative quaternion Hopfield neural network (CQHNN) as the analogy of QHNN. The multiplication of commutative quaternions is commutative and convenient, unlike that of quaternions. We compared the noise tolerance of CQHNNs and QHNNs by computer simulation, and discuss the simulation results from the perspective of rotational invariance and self loops.

Almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks of neutral type with time delays in the leakage term

ABSTRACTIn this paper, we propose a class of quaternion-valued shunting inhibitory cellular neural networks of neutral type with time delays in the leakage term. In order to overcome the difficulty of the non-commutativity of quaternion multiplication, we first decompose the system under consideration into four real-valued systems. Then by using the exponential dichotomy theory of linear differential equations and Banach's fixed point theorem, we establish sufficient conditions to ensure the existence and global exponential stability of almost periodic solutions for this class of neural networks. Finally, a numerical example is given to demonstrate the effectiveness of the obtained result.

Why and How to Avoid the Flipped Quaternion Multiplication

Over the last decades quaternions have become a crucial and very successful tool for attitude representation in robotics and aerospace. However, there is a major problem that is continuously causing trouble in practice when it comes to exchanging formulas or implementations: there are two quaternion multiplications commonly in use, Hamilton’s multiplication and its flipped version, which is often associated with NASA’s Jet Propulsion Laboratory. This paper explains the underlying problem for the popular passive world-to-body usage of rotation quaternions, and promotes an alternative solution compatible with Hamilton’s multiplication. Furthermore, it argues for discontinuing the flipped multiplication. Additionally, it provides recipes for efficiently detecting relevant conventions and migrating formulas or algorithms between them.

Global Mittag-Leffler stability and synchronization analysis of fractional-order quaternion-valued neural networks with linear threshold neurons

This paper talks about the stability and synchronization problems of fractional-order quaternion-valued neural networks (FQVNNs) with linear threshold neurons. On account of the non-commutativity of quaternion multiplication resulting from Hamilton rules, the FQVNN models are separated into four real-valued neural network (RVNN) models. Consequently, the dynamic analysis of FQVNNs can be realized by investigating the real-valued ones. Based on the method of M-matrix, the existence and uniqueness of the equilibrium point of the FQVNNs are obtained without detailed proof. Afterwards, several sufficient criteria ensuring the global Mittag-Leffler stability for the unique equilibrium point of the FQVNNs are derived by applying the Lyapunov direct method, the theory of fractional differential equation, the theory of matrix eigenvalue, and some inequality techniques. In the meanwhile, global Mittag-Leffler synchronization for the drive–response models of the addressed FQVNNs are investigated explicitly. Finally, simulation examples are designed to verify the feasibility and availability of the theoretical results.

Almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks with time-varying and distributed delays

AbstractIn this paper, we consider the problem of the almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks (QVHHNNs) with time-varying and distributed delays. Firstly, to avoid the non-commutativity of quaternion multiplication, we decompose QVHHNNs into an equivalent real-valued system. Secondly, we use the Banach fixed point theorem to obtain the existence of almost automorphic solutions of QVHHNNs. Thirdly, by designing a novel state-feedback controller and constructing suitable Lyapunov functions, we obtain that the drive-response structure of QVHHNNs with almost automorphic coefficients can realize the exponential synchronization. Our results are completely new. Finally, a numerical example is given to illustrate the feasibility of our results.

Versatile Quaternion Multipliers Based on Distributed Arithmetic

This paper introduces the idea of versatile circuits for multiplying 4-dimensional hypercomplex numbers in hardware. Depending on the settings of such a device, a variable quaternion can be left- or right-multiplied by a constant coefficient or by its conjugate, as various operations are useful in transform-type algorithms. Multiplierless circuits based on distributed arithmetic (DA) are reviewed that compute quaternion products by additions and bit shifts. It is shown that they can be made versatile by extending memory of partial results, but a better solution is our method for preprocessing bits used to address this memory. The method allows for using the same partial results to compute different inner products that are related to a quaternion multiplication. So versatile multipliers can be implemented with memory optimized so as to save area or to speed up reprogramming compared to the basic DA-based circuit. This has been demonstrated by hardware design experiments, which show that 13–69% area can be saved in a case of ASIC implementation, while for FPGA implementation, spending only 11% more logic resources allows a multiplier to be reprogrammed 75% faster. Additionally, it has been explained how versatile multipliers can be used to realize low-area analysis/synthesis filter banks.

Global µ-stability of quaternion-valued neural networks with mixed time-varying delays

In this paper, the problem of global µ-stability for quaternion-valued neural networks with time-varying delays and unbounded distributed delays is investigated. To avoid the non-commutativity of quaternion multiplication, the quaternion-valued neural networks is decomposed into two complex-valued systems. By employing the homomorphic mapping principle, a sufficient condition for the existence and uniqueness of equilibrium point of the considered quaternion-valued neural networks is proposed in the form of linear matrix inequality (LMI) in complex-valued domain. Further, the appropriate Lyapunov–Krasovkii functional is constructed in the Hermitian quadratic form, and sufficient condition to ensure the global µ-stability of the equilibrium point is obtained by using inequality technique. Finally, two numerical examples with simulations are provided to verify the effectiveness of the obtained results.

Using Dual Quaternion to Study Translational Surfaces

A translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces can also be generated from two rational space curves by dual quaternion multiplication. Using the mathematics of dual quaternions, we provide a necessary and sufficient condition for a rational tensor product surface to be a translational surface. Examples are provided to illustrate our theorems and flesh out our algorithms.

Stability Analysis of Quaternion-Valued Neural Networks: Decomposition and Direct Approaches.

In this paper, we investigate the global stability of quaternion-valued neural networks (QVNNs) with time-varying delays. On one hand, in order to avoid the noncommutativity of quaternion multiplication, the QVNN is decomposed into four real-valued systems based on Hamilton rules: $ij=-ji=k,~jk=-kj=i$ , $ki=-ik=j$ , $i^{2}=j^{2}=k^{2}=ijk=-1$ . With the Lyapunov function method, some criteria are, respectively, presented to ensure the global $\mu $ -stability and power stability of the delayed QVNN. On the other hand, by considering the noncommutativity of quaternion multiplication and time-varying delays, the QVNN is investigated directly by the techniques of the Lyapunov-Krasovskii functional and the linear matrix inequality (LMI) where quaternion self-conjugate matrices and quaternion positive definite matrices are used. Some new sufficient conditions in the form of quaternion-valued LMI are, respectively, established for the global $\mu $ -stability and exponential stability of the considered QVNN. Besides, some assumptions are presented for the two different methods, which can help to choose quaternion-valued activation functions. Finally, two numerical examples are given to show the feasibility and the effectiveness of the main results.

On Algebras of Three-Dimensional Quaternion Harmonic Fields

A quaternion field is a pair p = {α, u} of a function α and a vector field u given on a 3d Riemannian manifold Ω with boundary. A field is said to be harmonic if ∇α = rot u in Ω. The linear space of harmonic fields is not an algebra with respect to quaternion multiplication. However, it may contain commutative algebras, which is the subject of the paper. Possible applications of these algebras to the impedance tomography problem are touched upon.

Decomposition approach to the stability of recurrent neural networks with asynchronous time delays in quaternion field

In this paper, the global exponential stability for recurrent neural networks (QVNNs) with asynchronous time delays is investigated in quaternion field. Due to the non-commutativity of quaternion multiplication resulting from Hamilton rules: ij=−ji=k, jk=−kj=i, ki=−ik=j, ijk=i2=j2=k2=−1, the QVNN is decomposed into four real-valued systems, which are studied separately. The exponential convergence is proved directly accompanied with the existence and uniqueness of the equilibrium point to the consider systems. Combining with the generalized ∞-norm and Cauchy convergence property in the quaternion field, some sufficient conditions to guarantee the stability are established without using any Lyapunov–Krasovskii functional and linear matrix inequality. Finally, a numerical example is given to demonstrate the effectiveness of the results.

Rapid convergence of optimal control in NMR using numerically-constructed toggling frames

We present a numerical method for rapidly solving the Bloch equation for an arbitrary time-varying spin-1/2 Hamiltonian. The method relies on fast, vectorized computations such as summation and quaternion multiplication, rather than slow computations such as matrix exponentiation. A toggling frame is constructed in which the Hamiltonian is time-invariant, and therefore has a simple analytical solution. The key insight is that constructing this frame is faster than solving the system dynamics in the original frame. Rapidly solving the Bloch equations for an arbitrary Hamiltonian is particularly useful in the context of NMR optimal control. Optimal control theory can be used to design pulse shapes for a range of tasks in NMR spectroscopy. However, it requires multiple simulations of the Bloch equations at each stage of the algorithm, and for each relevant set of parameters (e.g. chemical shift frequencies). This is typically time consuming. We demonstrate that by working in an appropriate toggling frame, optimal control pulses can be generated much faster. We present a new alternative to the well-known GRAPE algorithm to continuously update the toggling-frame as the optimal pulse is generated, and demonstrate that this approach is extremely fast. The use and benefit of rapid optimal pulse generation is demonstrated for 19F fragment screening experiments.