Obtaining Triplet from Quaternions

In this study, we obtain triplets from quaternions. First, we obtain triplets from real quaternions. Then, as an application of this, we obtain dual triplets from the dual quaternions. Quaternions, in many areas, it allows ease in calculations and geometric representation. Quaternions are four dimensions. The triplets are in three dimensions. When we express quaternions with triplets, our work is made even easier. Quaternions are very important in the display of rotational movements. Dual quaternions are important in the expression of screw movements. Reducing movements from four dimensions to three dimensions makes our work easier. This simplicity is achieved by obtaining triplets from quaternions.

Some Characterizations for a Quaternion-Valued and Dual Variable Curve

Quaternions, which are found in many fields, have been studied for a long time. The interest in dual quaternions has also increased after real quaternions. Nagaraj and Bharathi developed the basic theories of these studies. The Serret–Frenet Formulae for dual quaternion-valued functions of one real variable are derived. In this paper, by making use of the results of some previous studies, helixes and harmonic curvature concepts in Q D 3 and Q D 4 are considered and a characterization for a dual harmonic curve to be a helix is given.

TEST MAP AND DISCRETENESS IN SL(2, ℍ)

Let ℍ be the division ring of real quaternions. Let SL(2, ℍ) be the group of 2 × 2 quaternionic matrices $A={\scriptsize{(\begin{array}{l@{\quad}l} a & b \\ c & d \end{array})}}$ with quaternionic determinant det A = |ad − aca−1b| = 1. This group acts by the orientation-preserving isometries of the five-dimensional real hyperbolic space. We obtain discreteness criteria for Zariski-dense subgroups of SL(2, ℍ).

Byte weight enumerators and modular forms of genus r

For a positive integer m, let R be either the ring ℤ2m of integers modulo 2m or the quaternionic ring Σ2m = ℤ2m + αℤ2m + βℤ2m + γℤ2m with α = 1 + î, β = 1 + ĵ and [Formula: see text], where [Formula: see text] are elements of the ring ℍ of real quaternions satisfying [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we obtain Jacobi forms (or Siegel modular forms) of genus r from byte weight enumerators (or symmetrized byte weight enumerators) in genus r of Type I and Type II codes over R. Furthermore, we derive a functional equation for partial Epstein zeta functions, which are summands of classical Epstein zeta functions associated with quadratic forms.

Lagrange polynomials over Clifford numbers

We construct Lagrange interpolating polynomials for a set of points and values belonging to the algebra of real quaternions ℍ ≃ ℝ0,2, or to the real Clifford algebra ℝ0,3. In the quaternionic case, the approach by means of Lagrange polynomials is new, and gives a complete solution of the interpolation problem. In the case of ℝ0,3, such a problem is dealt with here for the first time. Elements of the recent theory of slice regular functions are used. Leaving apart the classical cases ℝ0,0 ≃ ℝ, ℝ0,1 ≃ ℂ and the trivial case ℝ1,0 ≃ ℝ⊕ℝ, the interpolation problem on Clifford algebras ℝp,q with (p,q) ≠ (0,2), (0,3) seems to have some intrinsic difficulties.

Completing a2×2Block Matrix of Real Quaternions with a Partial Specified Inverse

This paper considers a completion problem of a nonsingular2×2block matrix over the real quaternion algebraℍ: Letm1, m2, n1, n2be nonnegative integers,m1+m2=n1+n2=n>0, andA12∈ℍm1×n2, A21∈ℍm2×n1, A22∈ℍm2×n2, B11∈ℍn1×m1be given. We determine necessary and sufficient conditions so that there exists a variant block entry matrixA11∈ℍm1×n1such thatA=(A11A12A21A22)∈ℍn×nis nonsingular, andB11is the upper left block of a partitioning ofA-1. The general expression forA11is also obtained. Finally, a numerical example is presented to verify the theoretical findings.