Quaternion Multiplication Research Articles

Galois action on $\bar{\mathbb Q}$-isogeny classes of abelian $L$-surfaces with quaternionic multiplication

An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the Galois action on the endomorphism rings. We can think of them as generalizations of abelian varieties defined over L with endomorphisms also defined over L. In the one dimensional case, an elliptic curve defined over L gives rise to a Galois representation provided by the Galois action on its Tate module. This classical Galois representation has been a central object of study in Number Theory over the last decades. Besides, given an elliptic L-curve one can construct a projective analogue of the previous Galois representation. In this work we construct similar projective representations in the two-dimensional case, namely, attached to abelian L-surfaces with quaternionic multiplication or fake elliptic curves. Moreover, we prove that such projective representations also describe the Galois action on the isogeny class of the elliptic L-curve, in the classical case, or the abelian L-surface with quaternionic multiplication, in the general case.

Visualizing Quaternion Multiplication

Quaternion rotation is a powerful tool for rotating vectors in 3-D; as a result, it has been used in various engineering fields, such as navigations, robotics, and computer graphics. However, understanding it geometrically remains challenging, because it requires visualizing 4-D spaces, which makes exploiting its physical meaning intractable. In this paper, we provide a new geometric interpretation of quaternion multiplication using a movable 3-D space model, which is useful for describing quaternion algebra in a visual way. By interpreting the axis for the scalar part of quaternion as a 1-D translation axis of 3-D vector space, we visualize quaternion multiplication and describe it as a combined effect of translation, scaling, and rotation of a 3-D vector space. We then present how quaternion rotation formulas and the derivative of quaternions can be formulated and described under the proposed approach.

Icosahedral invariants and Shimura curves

Shimura curves are moduli spaces of abelian surfaces with quaternion multiplication. Models of Shimura curves are very important in number theory. Klein's icosahedral invariants $\mathfrak{A},\mathfrak{B}$ and $\mathfrak{C}$ give the Hilbert modular forms for $\sqrt{5}$ via the period mapping for a family of $K3$ surfaces. Using the period mappings for several families of $K3$ surfaces, we obtain explicit models of Shimura curves with small discriminant in the weighted projective space ${\rm Proj} (\mathbb{C}[\mathfrak{A},\mathfrak{B},\mathfrak{C}])$.

Existence and Global Exponential Stability of Pseudo Almost Periodic Solutions for Neutral Type Quaternion-Valued Neural Networks with Delays in the Leakage Term on Time Scales

We propose a class of neutral type quaternion-valued neural networks with delays in the leakage term on time scales that can unify the discrete-time and the continuous-time neural networks. In order to avoid the difficulty brought by the noncommutativity of quaternion multiplication, we first decompose the quaternion-valued system into four real-valued systems. Then, by applying the exponential dichotomic theory of linear dynamic equations on time scales, Banach’s fixed point theorem, the theory of calculus on time scales, and inequality techniques, we obtain some sufficient conditions on the existence and global exponential stability of pseudo almost periodic solutions for this class of neural networks. Our results are completely new even for both the case of the neural networks governed by differential equations and the case of the neural networks governed by difference equations and show that, under a simple condition, the continuous-time quaternion-valued network and its corresponding discrete-time quaternion-valued network have the same dynamical behavior for the pseudo almost periodicity. Finally, a numerical example is given to illustrate the feasibility of our results.

On the local–global divisibility of torsion points on elliptic curves and GL2-type varieties

Let p be a prime number and let k be a number field. Let E be an elliptic curve defined over k. We prove that if p is odd, then the local–global divisibility by any power of p holds for the torsion points of E. We also show with an example that the hypothesis over p is necessary.We get a weak generalization of the result on elliptic curves to the larger family of GL2-type varieties over k. In the special case of the abelian surfaces A/k with quaternionic multiplication over k we obtain that for all prime numbers p, except a finite number depending only on the isomorphism class of the ring Endk(A), the local–global divisibility by any power of p holds for the torsion points of A.

Global μ-stability criteria for quaternion-valued neural networks with unbounded time-varying delays

In this paper, we first propose quaternion-valued neural networks (QVNNs) with unbounded time-varying delays. Some sufficient conditions on the global μ-stability in the form of both complex-valued and real-valued linear matrix inequalities (LMIs) are provided by solving two difficulties. One is decomposing the QVNN into two complex-valued systems with the plural decomposition method of quaternion, which can reduce the complexity of calculations by avoiding the non-commutativity of quaternion multiplication. The other is choosing the appropriate Lyapunov–Krasovskii functional in the form of Hermitian matrices, which is a big challenge. Finally, two numerical examples are provided to verify the effectiveness of the obtained results.

DUAL QUATERNIONIC REGULAR FUNCTION OF DUAL QUATERNION VARIABLES

We give representations of differential operators and rules for addition and multiplication of dual quaternions. Also, we research the notions and properties of a regular function and a corresponding harmonic function with values in dual quaternions of Clifford analysis.

Abelian Surfaces with an Automorphism and Quaternionic Multiplication

AbstractWe construct one-dimensional families of Abelian surfaces with quaternionic multiplication, which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)- polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space that parametrizes Abelian surfaces with real multiplication by .

On a new generalization of Fibonacci quaternions

In this paper, we present a new generalization of the Fibonacci quaternions that are emerged as a generalization of the best known quaternions in the literature, such as classical Fibonacci quaternions, Pell quaternions, k -Fibonacci quaternions. We give the generating function and the Binet formula for these quaternions. By using the Binet formula, we obtain some well-known results. Also, we correct some results in [3] and [4] which have been overlooked that the quaternion multiplication is non commutative.

SOME PROPERTIES OF MATRIX ALGEBRA OF SEMI-QUATERNIONS

Abstract. By representing semi-quaternions as four-dimensional vectors and the multiplication of quaternions as matrix-by-vector product, we investi- gate properties of matrix associated with a semi-quaternion and examine De- Moivre's formula for this matrix, from which the nth power of such a matrix can be determined.

Generalized Legendre curves and quaternionic multiplication

This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of 2-dimensional abelian varieties of this kind, we determine when the endomorphism algebra of each fiber defined over the algebraic closure of Q contains a quaternion algebra.

On the Semicircular Law of Large-Dimensional Random Quaternion Matrices

It is well known that the Gaussian symplectic ensemble is defined on the space of \(n\times n\) quaternion self-dual Hermitian matrices with Gaussian random elements. There is a huge body of literature regarding this kind of matrices based on the exact known form of the density function of the eigenvalues (see Erdős in Russ Math Surv 66(3):507–626, 2011; Erdős in Probab Theory Relat Fields 154(1–2):341–407, 2012; Erdős et al. in Adv Math 229(3):1435–1515, 2012; Knowles and Yin in Probab Theory Relat Fields, 155(3–4):543–582, 2013; Tao and Vu in Acta Math 206(1):127–204, 2011; Tao and Vu in Electron J Probab 16(77):2104–2121, 2011). Due to the fact that multiplication of quaternions is not commutative, few works about large-dimensional quaternion self-dual Hermitian matrices are seen without normality assumptions. As in natural, we shall get more universal results by removing the Gaussian condition. For the first step, in this paper, we prove that the empirical spectral distribution of the common quaternion self-dual Hermitian matrices tends to the semicircular law. The main tool to establish the universal result is given as a lemma in this paper as well.

Adaptive Filtering Using Complex Data and Quaternions

Abstract We provide an overview of complex-data and quaternion-based nonlinear adaptive filtering. The use of quaternion-valued data has been drawing recent interest in various areas of statistical signal processing, including adaptive filtering, image pattern recognition, and modeling and tracking of motion. Specifically, linear and widely linear adaptive filter designs based on quaternion data have been presented in the literature. The benefit for quaternion-valued processing in particular includes performing data transformations in 3 or 4-dimensional space conveniently compared to vector algebra. The transformations are performed using quaternion addition and multiplication, which differs from real and complex-valued multiplication in that quaternion multiplication is non-commutative. Recently, new algorithms have been developed that outperform others for these kinds of problems. We present an overview and discuss performance analysis results as well as simulations to verify the theory.

Low read‐only memory distributed arithmetic implementation of quaternion multiplier using split matrix approach

In most algorithms that use quaternion numbers, the key operation is a quaternion multiplication, of which the efficiency and accuracy obviously determine the same properties of the whole computational scheme of a filter or transform. A digit (L-bit)-serial quaternion multiplier based on the distributed arithmetic (DA) using the splitting of the multiplication matrix is presented. The circuit provides the facility to compute several products of quaternion components concurrently as well as to reduce the memory capacity by half in comparison with the known DA-based multiplier, and it is well suited for field programmable gate array (FPGA)-based fixed-point implementations of the algorithms. Apart from a theoretical development, the experimental design results which are obtained using a Xilinx Virtex 6 FPGA are reported.

Quaternion rational surfaces: Rational surfaces generated from the quaternion product of two rational space curves

A quaternion rational surface is a surface generated from two rational space curves by quaternion multiplication. The goal of this paper is to demonstrate how to apply syzygies to analyze quaternion rational surfaces. We show that we can easily construct three special syzygies for a quaternion rational surface from a μ-basis for one of the generating rational space curves. The implicit equation of any quaternion rational surface can be computed from these three special syzygies and inversion formulas for the non-singular points on quaternion rational surfaces can be constructed. Quaternion rational ruled surfaces are generated from the quaternion product of a straight line and a rational space curve. We investigate special μ-bases for quaternion rational ruled surfaces and use these special μ-bases to provide implicitization and inversion formulas for quaternion rational ruled surfaces. Finally, we show how to determine if a real rational surface is also a quaternion rational surface.

Modular Abelian Varieties Over Number Fields

AbstractThe main result of this paper is a characterization of the abelian varieties B/K defined over Galois number fields with the property that the L-function L(B/K; s) is a product of L-functions of non-CM newforms over ℚ for congruence subgroups of the form Γ1(N). The characterization involves the structure of End(B), isogenies between the Galois conjugates of B, and a Galois cohomology class attached to B/K.We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied, we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.

Galois representations with quaternion multiplication associated to noncongruence modular forms

In this paper we study the compatible family of degree-4 Scholl representations �` associated with a space S of weight � > 2 noncongruence cusp forms satisfying Quaternion Mul- tiplications over a biquadratic extension of Q. It is shown that �` is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for GL4 over Q. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in S and those of certain automorphic forms via the three-term Atkin and Swinnerton-Dyer congruences.

Two Sample Tests for Mean 3D Projective Shapes from Digital Camera Images

In this article, we extend mean 3D projective shape change in matched pairs to independent samples. We provide a brief introduction of projective shapes of spatial configurations obtained from their digital camera images, building on previous results of Crane and Patrangenaru (J Multivar Anal 102:225–237, 2011). The manifold of projective shapes of k-ads in 3D containing a projective frame at five given landmark indices has a natural Lie group structure, which is inherited from the quaternion multiplication. Here, given the small sample size, one estimates the mean 3D projective shape change in two populations, based on independent random samples of possibly different sizes using Efron’s nonparametric bootstrap. This methodology is applied in three relevant applications of analysis of 3D scenes from digital images: visual quality control, face recognition, and scene recognition.

Endomorphism algebras of QM abelian surfaces

We determine endomorphism algebras of abelian surfaces with quaternion multiplication.

Coordinated Multiple Spacecraft Attitude Control with Communication Time Delays and Uncertainties

In this paper, we consider the coordinated attitude control problem of spacecraft formation with communication delays, model and disturbance uncertainties, and propose novel synchronized control schemes. Since the attitude motion is essential in non-Euclidean space, thus, unlike the existing designs which describe the delayed relative attitude via linear algorithm, we treat the attitude error and the local relative attitude on the nonlinear manifold-Lie group, and attempt to obtain coupling attitude information by the natural quaternion multiplication. Our main focus is to address two problems: 1) Propose a coordinated attitude controller to achieve the synchronized attitude maneuver, i.e., synchronize multiple spacecraft attitudes and track a time-varying desired attitude; 2) With known model information, we achieve the synchronized attitude maneuver with disturbances under angular velocity constraints. Especially, if the formation does not have any uncertainties, the designer can simply set the controller via an appropriate choice of control gains to avoid system actuator saturation. Our controllers are proposed based on the Lyapunov-Krasovskii method and simulation of a spacecraft formation is conducted to demonstrate the effectiveness of theoretical results.