Narrow Results

Let ${𝒞}$ be a quaternion algebra or an octonion algebra over a field $F$ of characteristic $0$. In a previous work, we show that the $n$th symmetric power ${\mathrm{\text{Sym}}}^n{𝒞}$ of ${𝒞}$ is isomorphic to a direct sum of central simple algebras ${𝕊}_m^{(n)}{𝒞}$, $m=\lceil \frac{n}{2}\rceil ,\dots ,n$. In this work, we study the dual vector spaces of the ${𝕊}_m^{(n)}{𝒞}$’s. We show that ${({𝕊}_m^{(n)}{𝒞})}^{\ast }\cong {\mathrm{\text{Sym}}}^{2m-n}({{𝒞}}^{\ast }){\mu }^{n-m}/{\mathrm{\text{Sym}}}^{2m-n-2}({{𝒞}}^{\ast }){\mu }^{n-m+1}$, under the isomorphism induced by the isomorphism ${({{𝒞}}^{\otimes n})}^{\ast }\cong {({{𝒞}}^{\ast })}^{\otimes n}$, where $\mu \in {({{𝒞}}^{\ast })}^{\otimes 2}={({{𝒞}}^{\otimes 2})}^{\ast }$ is the $F$-linear map ${𝒞}\otimes {𝒞}\to F$ defined by $\mu (a\otimes b)=\mathrm{\text{Ntr}}(āb)$ for $a,b\in {𝒞}$, with $\mathrm{\text{Ntr}}$ and $\stackrel{̄}{}$ being the normalized trace and involution, respectively, of ${𝒞}$.

Ellipse motion involving an angle about a vector is referred to as elliptical rotation. The purpose of this work is to define a new class of elliptic quaternions, elliptic inner product, and elliptic vector product, which we refer to as multiplicative elliptic quaternions. Then, utilizing those multiplicative elliptic quaternions, we produce a multiplicative elliptical rotation matrix and provide several applications. In the end, the theorems established were presented along geometric and physical applications suitable for the quaternionic structure with the help of Maple. There were also given examples of various applications.

Biological evidence shows that there are neural networks specialized for recognition of signals and patterns acting as associative memories. The spiking neural networks are another kind which receive input from a broad range of other brain areas to produce output that selects particular cognitive or motor actions to perform. An important contribution of this work is to consider the geometric processing in the modeling of feed-forward neural networks. Since quaternions are well suited to represent 3D rotations, it is then well justified to extend real-valued neural networks to quaternion-valued neural networks for task of perception and control of robot manipulators. This work presents the quaternion spiking neural networks which are able to control robots, where the examples confirm that these artificial neurons have the capacity to adapt on-line the robot to reach the desired position. Also, we present the quaternionic quantum neural networks for pattern recognition using just one quaternion neuron. In the experimental analysis, we show the excellent performance of both quaternion neural networks.

This paper investigates some aspects of the arithmetic of a quintic threefold in Pr⁴ with double points singularities. Particular emphasis is given to the study of the L-function of the Galois action ρ on the middle ℓ-adic cohomology. The main result of the paper is the proof of the existence of a Hilbert modular form of weight (2, 4) and conductor 30, on the real quadratic field , whose associated (continuous system of) Galois representation(s) appears to be the most likely candidate to induce the scalar extension . The Hilbert modular form is interpreted as a common eigenvector of the Brandt matrices which describe the action of the Hecke operators on a space of theta series associated to the norm form of a quaternion algebra over and a related Eichler order.

Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as E³ (Euclidean 3-space), H³ (hyperbolic 3-space) and E^{2, 1} (Minkowski 3-space), using quaternion algebra theory, are studied. We study the different representations of a 2-generator group in which the generators are send to conjugate elements, by analyzing the points of an algebraic variety, that we call the variety of affine c-representations ofG. Each point in this variety corresponds to a representation in the unit group of a quaternion algebra and their affine deformations.

The complete classification of representations of the Trefoil knot group G in S³ and SL(2, ℝ), their affine deformations, and some geometric interpretations of the results, are given. Among other results, we also obtain the classification up to conjugacy of the noncyclic groups of affine Euclidean isometries generated by two isometries μ and ν such that μ² = ν³ = 1, in particular those which are crystallographic. We also prove that there are no affine crystallographic groups in the three-dimensional Minkowski space which are quotients of G.

One of the most often used techniques to represent color images is quaternion algebra. This study introduces the quaternion Krawtchouk moments, QKrMs, as a new set of moments to represent color images. Krawtchouk moments (KrMs) represent one type of discrete moments. QKrMs use traditional Krawtchouk moments of each color channel to describe color images. This new set of moments is defined by using orthogonal polynomials called the Krawtchouk polynomials. The stability against the translation, rotation, and scaling transformations for QKrMs is discussed. The performance of the proposed QKrMs is evaluated against other discrete quaternion moments for image reconstruction capability, toughness against various types of noise, invariance to similarity transformations, color face image recognition, and CPU elapsed times.

Let K be a p-adic local field where p is an odd prime and let A be the unique quaternion division algebra whose centre is K. By means of Stiefel–Whitney classes, we define an exponential homomorphism ϒ_K from the orthogonal representations of A*/K* to fourth roots of unity. We then evaluate this homomorphism in terms of the local root numbers of two-dimensional symplectic Galois representations of K, using the Langlands correspondence relating Galois representations to continuous representations of A*.

Let k be a field of characteristic distinct from 2, a, a₁, a₂, a₃ ∈ k*, D ∈ ₂Br k, exp D = 2, . We prove that D is a sum of 18 quaternion algebras. Also for a field F of certain type we construct a certain function f(ind D) such that D is a sum of f(ind D) quaternions for any D ∈ ₂Br (F).

In this paper, we present a solution for any standard quaternion quadratic equation, i.e. an equation of the form z² + μz + ν = 0 where μ and ν belong to some quaternion division algebra Q over some field F, assuming the characteristic of F is 2.

We show that if $I$ is a non-central Lie ideal of a ring $R$ with Char$(R)\ne 2$, such that all of its nonzero elements are invertible, then $R$ is a division ring. We prove that if $R$ is an $F$-central algebra and $I$ is a Lie ideal without zero divisor such that the set of multiplicative cosets $\{aF|a\in I\}$ is of finite cardinality, then either $R$ is a field or $I$ is central. We show the only non-central Lie ideal without zero divisor of a non-commutative central $F$-algebra $R$ with Char$(R)\ne 2$ and radical over the center is $[R,R]$, the additive commutator subgroup of $R$ and in this case $R$ is a generalized quaternion algebra. Finally we prove that if $I$ is a Lie ideal without zero divisor in a central $F$-algebra with characteristic not 2 and if $(\frac{I+F}{F},+)$ is a finite residual group, then $I$ is central.

In characteristic two, we determine all possible total decompositions of a totally decomposable algebra with orthogonal involution into tensor products of quaternion algebras with involution.

Let $p$ be a multilinear polynomial in several noncommuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is a vector space. In this paper, we settle the analogous conjecture for a quaternion algebra.

We study systems of quadratic forms over fields and their isotropy over $2$-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence we obtain that every central simple algebra of degree $16$ is split by a $2$-extension of degree at most $2^{16}$.

We give a process to construct non-split, three-dimensional simple Lie algebras from involutions of $𝔰𝔩(2,k)$, where $k$ is a field of characteristic not two. Up to equivalence, non-split three-dimensional simple Lie algebras obtained in this way are parametrized by a subgroup of the Brauer group of $k$ and are characterized by the fact that their Killing form represents $-2$. Over local and global fields we re-express this condition in terms of Hilbert and Legendre Symbols and give examples of three-dimensional simple Lie algebras which can and cannot be obtained by this construction over the field of rationals.

Let ${𝒞}$ be a composition algebra which is either the Hamilton quaternion algebra $ℍ$ or the Cayley octonion algebra $𝕆$ over $ℝ$. In a previous work, the nth symmetric power ${\mathrm{\text{Sym}}}^n{𝒞}$ of ${𝒞}$ is shown to be a direct sum of central simple algebras, corresponding to the partitions of $n$ of length $2$, such that the component corresponding to the partition $(m,n-m)$ is isomorphic to the component $T_{2m-n}{𝒞}$ of ${\mathrm{\text{Sym}}}^{2m-n}{𝒞}$ corresponding to the partition $(2m-n,0)$ of $2m-n$. In this work, we study the building blocks $T_n{𝒞}$ of these decompositions. We show that the “local” structure of ${𝒞}$, i.e. the complex-like subfields of ${𝒞}$, determine both the complement of $T_n{𝒞}$ in ${\mathrm{\text{Sym}}}^n{𝒞}$ and the trace map of $T_n{𝒞}$, induced from the trace map of ${𝒞}$. We also derive a recursive trace formula on the $T_n{𝒞}$’s. We use the “local-global” results to define positive definite symmetric bilinear forms on the vector space ${\oplus }_{n=0}^{\infty }{T}_n{𝒞}$, which has a natural structure of a commutative and associative algebra. Finally, the structure of the central simple algebra $T_n{𝒞}$ is described.

A two-dimensional (2D) quaternion Fourier transform (QFT) defined with the kernel is proposed. Some fundamental properties, such as convolution, Plancherel and vector differential theorems, are established. The heat equation in quaternion algebra is presented as an example of the application of the QFT to partial differential equations. The wavelet transform is extended to quaternion algebra using the kernel of the QFT.

Detail enhancement of color images is required in many applications. Unsharp masking (UM) is the most classical tool for detail enhancement. Many generalizing UM approaches have been proposed, for example, the rational UM technique, the cubic unsharp technique, the adaptive UM technique and so on. For color images, these algorithms have three steps: (a) Implement the color2gray step; (b) design an extracting method of high frequency information (HFI) based on the luminance component (LC); (c) complete the enhancing process utilizing the HFI. However, using only the HFI of the LC may lose the HFI of the chrominance component (CC). This paper proposes a quaternion based detail enhancement algorithm to extract details of the color image using both of the luminance and CCs. The proposed algorithm is designed to address three tasks: (1) designing an extraction method of the color high frequency information (CHFI) based on quaternion description of the 3D vector rotation; (2) performing an effective fusion strategy of the CHFI and the gray high frequency information (GHFI); (3) designing a quaternion based measure method of the local dynamic range, based on which the enhancement coefficients of the proposed algorithm can be determined. The performance of the proposed algorithm compares favorably with many other similar enhancement algorithms. The eight parameters can be adjusted to control the sharpness to produce the desired results, which makes the proposed algorithm practically useful.

Since the introduction of quaternion by Hamilton in 1843, quaternions have been used in a lot of applications. One of the most interesting qualities is that we can use quaternions to carry out rotations and operate on other quaternions; this characteristic of the quaternions inspired us to investigate how the quantum states and quantum operator work in the field of quaternions and how we can use it to construct a quantum neural network. This new type of quantum neural network (QNN) is developed in the quaternion algebra framework that is isomorphic to the rotor algebra $G_3^{+}$ of the geometric algebra and is based on the so-called qubit neuron model. The quaternion quantum neural network (QQNN) is tested and shows robust performance.

In this paper, we introduce generalized-quaternionic Kähler analogue of Lagrangian and Hamiltonian mechanical systems. Finally, the geometrical-physical results related to generalized-quaternionic Kähler mechanical systems are also given.