Extensions of real numbers in more than two dimensions, in particular quaternions and octonions, are finding applications in physics due to the fact that they naturally capture symmetries of physical systems. However, in the conventional mathematical construction of complex and multicomplex numbers multiplication rules are postulated instead of being derived from a general principle. A more tra...

Hilbert’s law of quadratic reciprocity, as reformulated by Hasse, states that for a quaternion algebra D over a number field K, the number of ramified places of D is even. In general a number field has three kinds of places: finite, real, and complex. The complex places are never ramified. If D is assumed unramified at all finite places, it follows that D is ramified at an even number of real p...

In this paper, we propose a nonlinear adaptive position and attitude tracking controller for a rigid body that requires no information about the mass and inertia matrix of the body. Moreover, we provide sufficient conditions on the reference trajectory that guarantee mass and inertia matrix identification. The controller is shown to be almost globally asymptotically stable and can handle large ...

The functions studied in the paper are quaternion-valued of a quaternionic variable. It is shown that left slice regular and right related by particular involution, intrinsic play central role theory functions. relation between functions, revealed. As an application, classical Laplace transform generalized naturally to quaternions two different ways, which function real variable or function. us...

The main objective of this article is to give a survey on elementary functions in the context of quaternionic analysis. We define some of their more common properties, which as in the real and complex cases, will be familiar to the reader. This leads to the consideration of quaternion-valued functions depending on a quaternion variable, that is, functions whose input and output are quaternions....

Interior point methods can be extended to a number of cones (self-dual homogeneous cones) • Rn (linear programming) • vectorized symmetric matrices over real numbers (semidefinite programming) • vectorized Hermitian matrices over complex numbers • vectorized Hermitian matrices over quaternions • vectorized Hermitian 3×3 matrices over octonions Grötschel, Lovász and Schrijver [3]: semidefinite p...

As we know, quaternions are very efficient for representing rotations with clear geometric meaning (rotation axis and angle) and only one redundancy. Unfortunately, they do not handle translations, which meanwhile can be made multiplicative along with rotations via the use of homogeneous coordinates. Despite also being 4-tuples, homogeneous coordinates are algebraically incompatible with quater...

We present a practical and efficient means to compute the singular value decomposition (svd) of a quaternion matrix A based on bidiagonalization of A to a real bidiagonal matrix B using quaternionic Householder transformations. Computation of the svd of B using an existing subroutine library such as lapack provides the singular values of A. The singular vectors of A are obtained trivially from ...

Let [Formula: see text] be a field and finite-dimensional central division algebra over text]. We prove variant of the Nullstellensatz for text]-sided ideals in ring polynomial maps In case where is commutative, our main result reduces to text]-Nullstellensatz Laksov Adkins–Gianni–Tognoli. case, real numbers Hamilton quaternions, it quaternionic recently proved by Alon Paran.

The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective geometry of the weakly modular orthocomplemented lattice of propositions may be imbedded in a complex Hilbert space; this is the structure which has traditiona...