skew field
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AbstractWe consider bivariate polynomials over the skew field of quaternions, where the indeterminates commute with all coefficients and with each other. We analyze existence of univariate factorizations, that is, factorizations with univariate linear factors. A necessary condition for existence of univariate factorizations is factorization of the norm polynomial into a product of univariate polynomials. This condition is, however, not sufficient. Our central result states that univariate factorizations exist after multiplication with a suitable univariate real polynomial as long as the necessary factorization condition is fulfilled. We present an algorithm for computing this real polynomial and a corresponding univariate factorization. If a univariate factorization of the original polynomial exists, a suitable input of the algorithm produces a constant multiplication factor, thus giving an a posteriori condition for existence of univariate factorizations. Some factorizations obtained in this way are of interest in mechanism science. We present an example of a curious closed-loop mechanism with eight revolute joints.

AbstractWe consider the technique of lifting frames to higher dimensions with the ridge idea that originally was introduced by Grafakos and Sansing. We pursue a novel approach with regard to a non-commutative setting, concretely the skew-field of quaternions. Moreover, we allow for splitting dimensions and for lifting with regard to multi-ridges. To this end, we introduce quaternionic Sobolev spaces and prove the corresponding embedding theorems. We mention as concrete examples quaternionic wavelet frames and quaternionic shearlet frames, and give the respective lifted families.

Abstract We focus on the description of the automorphism group Γ∥ of a Clifford-like parallelism ∥ on a 3-dimensional projective double space (ℙ(HF ), ∥ ℓ , ∥ r ) over a quaternion skew field H (of any characteristic). We compare Γ∥ with the automorphism group Γ ℓ of the left parallelism ∥ ℓ , which is strictly related to Aut(H). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose ∥ in such a way that Γ∥ is either properly contained in Γ ℓ or coincides with Γ ℓ even though ∥ ≠ ∥ ℓ .

In this paper, we define the notions of i-octahedron groupoid and i-OLI [resp., i-ORI and i-OI], and study some of their properties and give some examples. Also we deal with some properties for the image and the preimage of i-octahedron groupoids [resp., i-OLI, i-ORI and i-OI] under a groupoid homomorphism. Next, we introduce the concepts of i-octahedron subgroup and normal subgroup of a group and investigate some of their properties. In particular, we obtain a characterization of an i-octahedron subgroup of a group. Finally, we define an i-octahedron subring [resp., i-OLI, i-ORI and i-OI] of a ring and find some of their properties. In particular, we obtain two characterizations of i-OLI [resp., i-ORI and i-OI] of a ring and a skew field, respectively.

Abstract Quantum analogues of sets are defined by two simple assumptions, allowing enumeration, reminiscent of the Gram–Schmidt orthogonalization process. It is shown that any symmetric quantum set is a classical set of irreducible components, and that each irreducible component of size $>3$ is representable by an orthomodular space over a skew field with involution. For finite or sufficiently large irreducible components, invariance of quantum cardinality is proved. Topological quantum sets are introduced as quantum analogues of topological spaces; irreducible ones of size $>3$ are shown to be representable by Hilbert spaces over ${\mathbb{R}}$, ${\mathbb{C}}$, or ${\mathbb{H}}$. Symmetric quantum sets are characterized as a class of $L$-algebras with an intrinsic geometry, and they are shown to be equivalent to Piron’s quantum formalism. Equivalences between symmetric quantum sets and several other structures are established. To any symmetric quantum set, a group with a right invariant lattice structure is associated as a complete invariant. A simple and self-contained proof of Solèr’s theorem is included, which is used to prove that sufficiently large irreducible symmetric quantum sets come from a classical Hilbert space.

AbstractLet D be a division ring of fractions of a crossed product {F[G,\eta,\alpha]}, where F is a skew field and G is a group with Conradian left-order {\leq}. For D we introduce the notion of freeness with respect to {\leq} and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space {F((G))} of all formal power series in G over F with respect to {\leq}. From this we obtain that all division rings of fractions of {F[G,\eta,\alpha]} which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, {F[G,\eta,\alpha]} possesses a division ring of fraction which is free in this sense if and only if the rational closure of {F[G,\eta,\alpha]} in the endomorphism ring of the corresponding right F-vector space {F((G))} is a skew field.

Abstract We are studying hyperbolic function theory in the total skew-field of quaternions. Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables. Our functions are depending on all four coordinates of quaternions. We consider functions, called $$\alpha $$α-hyperbolic harmonic, that are harmonic with respect to the Riemannian metric $$\begin{aligned} ds_{\alpha }^{2}=\frac{dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}{x_{3}^{\alpha }} \end{aligned}$$dsα2=dx02+dx12+dx22+dx32x3αin the upper half space $${\mathbb {R}}_{+}^{4}=\{\left( x_{0},x_{1},x_{2} ,x_{3}\right) \in {\mathbb {R}}^{4}:x_{3}>0\}$$R+4={x0,x1,x2,x3∈R4:x3>0}. If $$\alpha =2$$α=2, the metric is the hyperbolic metric of the Poincaré upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function $$x^{m}\,(m\in {\mathbb {Z}})$$xm(m∈Z), is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. Using fundamental $$\alpha $$α-hyperbolic harmonic functions, depending only on the hyperbolic distance and $$x_{3}$$x3, we verify a Cauchy type integral formula for conjugate gradient of $$\alpha $$α-hyperbolic harmonic functions. We also compare these results with the properties of paravector valued $$\alpha $$α-hypermonogenic in the Clifford algebra $${{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}$$Cℓ0,3.

In this paper, further results on the Drazin inverse are obtained in a ring. Several representations of the Drazin inverse of [Formula: see text] block matrices over an arbitrary ring are given under new conditions. Also, upper bounds for the Drazin index of block matrices are studied. Numerical examples are given to illustrate our results. Necessary and sufficient conditions for the existence as well as the expression of the group inverse of block matrices are obtained under certain conditions. In particular, some results of related papers which were considered for complex matrices, operator matrices and matrices over a skew field are extended to more general setting.